EVALUATION OF CERTAIN INFINITE SERIES USING ......uation of other particular infinite series,...

EVALUATION OF CERTAIN

INFINITE SERIES

USING THEOREMS

OF JOHN, RADEMACHER

AND KRONECKER

by

Colette Sharon Haley

B .A . (Honours), Carleton University

A thesis subm itted to

the Faculty of Graduate Studies and Research

in pa rtia l fu lfilm ent of

the requirements fo r the degree of

Master of Science

School of Mathematics and Statistics

Ottawa-Carleton In s titu te for Mathematics and Statistics

Carleton University

Ottawa, Ontario, Canada

© C opyright

2004

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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A bstract

John’s theorem relates the sum o f a certain in fin ite series involving a real

valued periodic function of period 1, which is o f bounded variation on [0, 1],

to a Riemann integral. A detailed proof of th is theorem based on the proof

o f Rademacher is given. John’s theorem is then used to determine the sum

o f some interesting in fin ite series. For example i t is shown th a t the sum of

1 _ 1 2 _ 2 2 _ 2 3 _ 3 £ _ _ 3 _ _3___3_ _ ± _ ±2 3 + 4 5 + 6 7 + 8 9 + 10 11+ 12 13+ 14 15+ 16 17 + " '

is Euler’s constant, see Theorem 1.6.2. Rademacher’s theorem is the exten

sion o f John’s theorem to algebraic number fields and th is theorem is applied

to determine the sum o f fu rther in fin ite series such as

1 1 1 2 1 1 2 7r log 21 2 4 + 5 8 + 9 10 + 4 ’

see Theorem 2.4.2. F inally, Kronecker’s lim it form ula is used to determine

the sums o f the in fin ite series

“ ( - 1)™ ~ ( - 1 )n “ ( - 1)™+"am? + bmn + cn? ’ ^ am2 + bmn + cn? ’ am2 + bmn + cn? ’

77i,n=—oo m ,n = —oo m ,n = —oo(m ,n )^ (0 ,0 ) (m ,n )^ (0 ,0 ) (m ,n ) / ( 0 ,0 )

fo r any prim itive , positive-definite, integral, b inary quadratic form

ax2 + bxy + cy2.

i


Acknowledgem ents

F irs t and foremost, I would like to thank my supervisor D r. Kenneth S.

W illiam s. He helped me to choose a topic and has guided me through the

d ifficu lt task of w riting a thesis. He has also been remarkably patient and

understanding, and for th is I thank him.

I would like to acknowledge the help and support from friends who have

encouraged me along the way, especially Chad Ternent, Tom Maloley and

M a tt Lemire.

I have dealt w ith many computer problems recently and would like to

thank everyone who has helped me to solve them.

This thesis was w ritten in George Gratzer’s book “M ath into

has been an extremely useful reference.

Lastly I am solely responsible for any errors and shortcomings le ft in th is

thesis.

ii


N otation

Z = domain o f integers

N = set of positive integers

Q = fie ld o f rational numbers

R = fie ld of real numbers

gcd(m, n) = greatest common divisor o f m, n 6 Z, (to, n) ^ (0,0)

( n 17 = Euler’s constant = 0.5772156649... = lim I V " ' - — log n

n —>oo \ ^ ' ?\ i= l

[rc] = greatest integer < x (x 6 R)

{ z } = fractional part o f x = x — [x] (x G R)

[a, b] = {x € R |a < x < b} (a, b € R, a < b )

{xo, X i, . . . , $„} = partition of [a, 6] given by a — xq < X\ < • • • < xn = b

A x k - x k - x k- \ (k = 1, 2 ,. . . , n)

p[a, 6] = set of all partitions of [o, 6]

= Legendre-Jacobi-Kronecker symbol (d € Z, d = 0,1 (mod 4) ,n

i i i

H (d ) = form class group o f discrim inant d

h(d) = form class number = order of form class group H (d)

f = conductor of discrim inant d

OO 00

5 3 / ( m»n) = 5 3m , n - —oo m ,n = —oo

(m ,n )^ (0 ,0 )


Contents

Abstract i

Acknowledgements ii

Notation iii

Introduction 1

1 John’s Theorem 5

1.1 Functions o f bounded variation .................................................. 5

1.2 John’s th e o re m ............................................................................... 17

1.3 P roof o f John’s th e o re m ............................................................... 23

1.4 Evaluation o f certain in fin ite series using John’s theorem . . . 36

1.5 The generalized Euler co n s ta n ts ...................................................... 41

v


CONTENTS vi

1.6 Dr. Vacca’s series for 7 ......................................................................49

2 Rademacher’s Theorem 56

2.1 N o ta tio n ............................................................................................... 56

2.2 Rademacher’s extension of John’s

th e o re m ................................................................................................ 61

2.3 Rademacher’s theorem for algebraic

number f ie ld s .......................................................................................62

2.4 Rademacher’s theorem for im aginary

quadratic f ie ld s ................................................................................ 63

2.4.1 RT = 0 (7 = 1 ), c = 1 + z .................................................... 65

2.4.2 t f = Q (7 = 2 ), c = 7 = 2 ........................................................69

2.4.3 K = Q {V = 7 ) ,c = 1 + f = 7 ..................................................72

2.5 Rademacher’s theorem for real quadratic fie ld s .............................75

2.5.1 K = Q (V2), c = 7 2 .......................................................... 77

2.5.2 K = Q(^/p), P (prime) = 3 (mod 4 ) ....................................... 79

2.6 Rademacher’s theorem for real cubic fields w ith two nonreal

em bedd ings.......................................................................................... 81

2.6.1 K = Q (7 2 ), c = 7 2 .......................................................... 84


CONTENTS vii

3 Kronecker’s Theorem 86

3.1 Dedekind eta function .................................................................... 86

3.2 Weber’s fu n c tio n s ............................................................................. 90

3.3 Kronecker’s lim it fo rm u la ................................................................ 91

3.4 F ina l re s u lts .........................................................................................103

Conclusion 109

Bibliography 110


Introduction

The aim of th is thesis is to use three theorems, namely John’s theorem,

Rademacher’s extension of John’s theorem, and Kronecker’s lim it formula,

to determine the sums o f certain in fin ite series.

In 1934 F ritz John [20] showed th a t i f f ( x ) is a real-valued function defined

for a ll real x such tha t

(1) f ( x ) is periodic of period 1,

(2) f ( x ) is o f bounded variation for 0 < x < 1,

Vand c — - is a rational number > 1 w ith q > 0 and gcd(p, q) = 1, then the Q

in fin ite series

1

INTRODUCTION 2

converges for a ll real values o f t, where

/

q, i i p ) ( n , q \ n ,

q - p , i f p | n, q [ n,

and its sum is

logc / f ( x ) dx.J o

Since f ( x ) is o f bounded variation on [0,1], f ( x ) is Riemann integrable on

[0,1], and so f * f ( x ) dx exists. In 1936 Hans Rademacher [27] proved John’s

theorem w ith (2) replaced by the weaker condition

(2)' f ( x ) is Riemann integrable on [0,1].

We give an exposition o f Rademacher’s proof o f John’s theorem expanding

on the details where necessary in Section 1.3.

Included in the volume [28] featuring his “lost” notebook are some frag

ments o f papers by Ramanujan. In particular, on pages 274 and 275 in [28],

there is the beginning o f a manuscript th a t probably was to focus on in te

grals related to Euler’s constant 7 . Berndt and Bowman [8] have presented

Ramanujan’s work in th is fragment and have related i t to other theorems in

the literature. In particu lar they prove [8, Lemma 2.5, p. 21] an integral

given by Ramanujan for Euler’s constant 7 , namely,

7 = /( 1 — xnn


INTRODUCTION 3

and use it to obtain (among others) a series representation of 7 due to Vacca

[33], namely

I t is our purpose to show th a t Vacca’s form ula for 7 , as well as the eval

uation o f other particu lar in fin ite series, follows easily from John’s theorem.

We just mention one such evaluation. We show in Section 1.5 how Liang

and Todd’s evaluation [24] o f the in fin ite series where k is

a positive integer, can be deduced easily from John’s theorem.

In his 1936 paper, Rademacher also extended John’s theorem to algebraic

number fields. In Chapter 2 we use Rademacher’s theorem to obtain the

evaluation o f certain in fin ite series involving binary quadratic forms. In the

case of positive-definite forms, we show for example tha t

Rademacher’s theorem. In the case of indefinite forms we prove for example

tha t

1<|m W |l<3+2>/2I 771—n v 2 I

In Chapter 3 we use Kronecker’s lim it form ula to evaluate the in fin ite

series

(m ,n )^ (0 ,0 )

where the sum is ordered by increasing values of m 2+ m n + 2n2, follows from

OO

m ,n = —0 0 m + n y /2 > 0

oo f 1 \m

(m,ra)^(0,0)

INTRODUCTION

E ( - i ) "am2 + bmn + cn21

771,71— —OO(m ,n )^ (0 ,0 )

^_-Qm+n

E am2 + bmn + cn2 ’771,71=—OO

(m ,n )# ( 0,0)

for any positive-definite, p rim itive , integral, binary quadratic form a

bxy + cy2. As an illus tra tion o f these, results, we show tha t

^ ( _ l ) m + n 47T ,

^ m 2 + 19n2 >/!9 g ’771,71=—OO v

(m ,n )^ (0 ,0 )

where 0 is the unique real root o f the cubic equation x3 — 2x — 2 = 0.


Chapter 1

John’s Theorem

1.1 Functions of bounded variation

John’s theorem, which is the main topic o f th is chapter, expresses an in fin ite

series involving a periodic function of bounded variation as a Riemann in

tegral. We begin by reviewing the basic properties of functions o f bounded

variation. We follow the treatm ent given by Apostol in [4, pp. 165-169], see

also [30, pp. 117-121].

Throughout th is section, a and b denote real numbers w ith a < b .

D e fin itio n 1.1.1. Let f be a real-valued function defined on the closed in

terval [a, 6]. If, fo r every pa ir o f points x and y in [a, b], x < y implies

f i x ) < f ( y ) , then f is said to be increasing on [a, b\. I f x < y implies

f ( x ) < f ( y ) then f is said to be strictly increasing on [a, 6]. Decreasing and

5

CHAPTER 1. JOHN’S THEOREM 6

strictly decreasing functions are sim ilarly defined.

Definition 1.1.2. A real-valued function f defined on the closed interval

[a, b] is said to be monotonic i f i t is increasing on [a, b] or decreasing on

[a,b].

Definition 1.1.3. I f [a,b] is a fin ite interval, then a fin ite set o f points

P = { x 0, x i , . . . , x n}

satisfying the inequalities a — Xq < Xi < • • ■ < xn- \ < xn — b is called a

partition o f \a,b]. The interval [xk- i , x k], k = 1,2, . . . , n is called the kth

subinterval o f P and we write Axk = Xk — %k-1> so that

n

Azfc = b — a.k = 1

The collection o f all possible partitions of [a,b ] is denoted by p[a, b}.

Definition 1.1.4. Let f be a real-valued function defined on [a, b]. I f P —

{ xq, x i , . . . , x n} is a partition o f [a,b], we set A f k = f ( x k ) — f { x k - 1)> k =

1 ,2 , . . . , n. I f there exists a positive number M such that

E ia /*i ^ Mk = l

fo r all partitions o f [a, b], then f is said to be o f bounded variation on [a, 6].

We now give some theorems that help us to decide when a function is of

bounded variation.

Theorem 1.1.1. Let f be a real-valued function defined on [a, 6]. I f f is

monotonic on [a, b], then f is o f bounded variation on [a, b].

Proof. As / is monotonic on [a, b], by D efin ition 1.1.2 / is either increasing

or decreasing on [a, b]. Suppose / increasing on [a, b\. Then, fo r any pa rtition

{x 0, x i , . . . , xn} o f [a, 6], we have f { x k) - i) > 0 giving A f k > 0. Now

53 ia ^ i = i t Afk = i t u ^ ) ~ = ~k = l k = 1 fe=l

so Y^k=i |A/fc| < M w ith M ~ f (b) — f (a) . Thus / is of bounded variation

on [a, 6]. I f / is decreasing on [a, 6] then —/ is increasing on [a, b] and the

above argument again shows th a t / is of bounded variation on [a, 6]. □

T heorem 1.1.2. Let f be a real-valued function defined on [a, b]. I f f is con

tinuous on [a, b] and i f f exists and is bounded in the interior, say \ f ( x ) \ < A

fo r all x in (a, b), then f is o f bounded variation on [a, 6],

Proof. Let { x q , x x, . . . , xn} be a p a rtition o f [a, 6], Applying the mean value

theorem to the subinterval [xk- X, xk], we have

A /* = f ( x k) - f ( x k- 1) = f ( t k) (xk - x k- x) = f ( t k) A x k

for some t k G (xk- X, x k). Then

n n n n

53 = 5 3 i/W A z fc i = 53 i/ '(4 ) iA£ft < ^ 5 3 a ®*.k—1 fc= l k = l k = l

Hence X X = i |A/fc| < M w ith M = A(b — a) so f is o f bounded variation on

[a, b]. □

The function f Ix2 cos—, x ± 0,

0, x = 0

is of bounded variation on [0,1] as / is continuous on [0,1] and

f sin — + 2x cos —, x ^ 0,m = \ x

^ 0, x = 0,

and

I/ '( z ) l < 3.

We note that f need not be bounded for / to be of bounded variation.

For example, f ( x ) = x 1/3 is monotonic so / is of bounded variation in every

finite interval (Theorem 1.1.1). However, f ' { x ) = ^73 —» +00 as x —» 0.

Not all continuous functions are of bounded variation. Consider the func

tion

{ a: sin ( —) , 0 < x < 2,\ x ) ’

0, x = 0,

which is continuous on [0,2], and the partition2 , U , 2

2n — S ’ 5 ’ 3

Here

*o = 0, x * = 2 n - ( 2 k - l ) ,k = 1,2, ' " , n ‘

For k = 2 , . . . , n we have,

A /* = f ( x k) - f ( x k- i )2 . ( (2n — (2k — l) ) 7r

sin 1 "2n — (2k — 1) \ 2

2 • ( (2n - (2k - 3) )n\2 n - ( 2 k - 3 ) V 2 /

9 9________ / 1 'jn —fe __________“ ______' / 0 7- 1 o v /2n — 2k -f-1 271 — 2k 3

/ 2 2( - 1) o ?T7- r - r +2n — 2k + 1 2n — 2& + 3

Hence

n n

E l A A l > e i a mk = l k = 2

2n - 2fc + 1 2n — 2k + 32 2

/ is not of bounded variation on [0, 2] as lim — = +oo.n—»+oo / K

T heo rem 1.1.3. Let f be a real-valued function on [a, 6]. I f f is o f bounded

variation on [a, 6], say X)|A/fc| < M fo r a ll partitions o f [a,b], then f is

bounded on [a,b]. In fact, \ f (x) \ < |/(a )| + M fo r x 6 [a, 6].

Proof. Let x € (a, b) and consider the p a rtition P = {a,x,b} . As / is of

bounded variation on [a, 6], we have

l / ( * ) l = I / O * ) “ / ( a ) + / ( a ) I < l / ( * ) “ / ( a ) I + l / ( a ) l < M + | / ( a ) |


|/ ( * ) - /(a ) | + | / ( 6) - / ( * ) | < M,

which gives | f ( x ) — f (a ) | < M . Now

as required.

If x = a, the inequality holds trivially. If x = b, the partition {a, b} gives

]/(&)| < 1/(6) - f ( a ) I + |/(o )| < M + \f(a)\. Hence |/(x )| < |/(a )| + M for

all x G [a, b}. □

The converse of Theorem 1.1.3 is not true. The function

{ 7Txs i n—, 0 < x < 2,

*0, x = 0,

is bounded on [0,2] as |/(x )| < |x| < 2 but is not of bounded variation on

[0,2].

Definition 1.1.5. Let f be a real-valued, function defined on [a,b]. Let f

be o f bounded variation on [a,b], and let J^(P ) denote the sum Y^k= i |A/fc|

corresponding to the partition P = {xo, x i , . . . , xn} o f [a, 6]. The number

V, = V /(a, b) = sup { Y ^ ( P )|P 6 p m }

is called the total variation o f f on the interval [a, 6].

Theorem 1.1.4. Let f and g be real-valued functions defined on [a,b]. Sup

pose fu rther that f and g are both o f bounded variation on [a, b]. Then so are

their sum, difference and product. Also, we have

vf±g < V f + Vg and Vfg < AVf + BVg,

where A = sup{|#(x)| |x G [a ,6]} and B = sup{]/(x)| |x G [a, 6]} .

Proof. We just prove the second of these two inequalities. Consider h(x) —

f (x)g(x) . Then for each p a rtition P o f [a, 6], we have

\Ahk\ = \ f ( xk)g(xk) - f ( x k - i ) g ( x k- i ) \

= If { x k)g{xk) - f { x k- x)g{xk)\

+ \ f ( x k- i ) g ( x k) - f ( x k- i )g(xk- i ) \

< A \A fk\ + B\Agk\.

Hence h = f g is o f bounded variation and V/g < AVf + BVa. □

There is a close relation between monotonic functions and functions of

bounded varition. We have already seen th a t monotonic functions are always

of bounded variation (Theorem 1.1.1), and we w ill see shortly th a t functions

of bounded variation can always be w ritten in terms o f monotonic functions

(Theorem 1.1.8). Being o f bounded variation is a stronger condition than

m onotonicity, indeed, the sum or product of monotonic functions need not

be monotonic. For example, x and —x2 are monotonic on [0,1], but x — x2

is not; and x is monotonic on [—1, 1] but x2 is not.

I t should be noted th a t the reciprocal of a function of bounded variation

is not necessarily of bounded variation. For instance, suppose f ( x ) —> 0 as

x —» c. Then 1 / / is not bounded in any interval containing c. Hence by

Theorem 1.1.3, 1 / / is not o f bounded variation on such an interval. Ex

cluding functions whose values get a rb itra rily close to zero le t us extend

Theorem 1.1.4 to quotients.

Theorem 1.1.5. Let f be o f bounded variation on [a,b] and assume that f

is bounded away from zero, that is there exists a positive number m such that

0 < m < \ f ( x ) \ fo r all x in [a, 6]. Then g = I f f is also o f bounded variation

on [a, b], and Vg < ^

Proof. We have

|A 0fc| =1 1 f ( x k- 1) - f { x k)) A f k

f(Xk) /(Zfc-l) f { x k) f ( x k- 1) f (X k ) f (x k- 1)< jA /fcl

m 2

□

Theorem 1.1.6. Let f be a real-valued function defined on [a, b]. Let f be o f

bounded variation on [a, 6], and assume that c € (a, 6). Then f is o f bounded

variation on [a, c] and on [c, b] and we have

Vf (a,b) = Vf (a,c) + Vf (c,b).

Proof. Consider partitions P i of [a,c] and P2 o f [c,b]. Then P0 = P i U P2 is

a pa rtition o f [a, b\. Denote by ]P (P ) the sum |A ./*| corresponding to a

p a rtition P . Then

J ^ iP l ) + = £ (i> „ ) < V,(a,b)-

Hence X X P i) and XX-^2) are bounded by Vf(a, b) so / is of bounded variation

on [a, c] and [c,b\. Now

£ ( P 0 + X ; « ! ) < V f M ) = * V,{a,c) + Vf ( c , b ) < V f (a,b).

For the reverse inequality, consider a pa rtition P = {xq,Xi , . . . , xn} <E

p[a, b] and Po = P U {c } the p a rtition obtained by adjoining the point c. I f

c 6 [£fc_i, xk\ then

l/(zfe) ~ f ( x k - 1)| < | f ( x k) - /(c)| + |/(c) - / ( z fe_i)|

giving X )(P ) ^ E M - The points of P0 in [a, c] give a p a rtitio n P i of [a, c]

and the points in [c, 6] give a pa rtition P2 o f [c, 6]. Now

E(p) s £«) = E(p>)+Ewo c)+*)•This shows th a t Vf(a, c) + Vf(c,b) is an upper bound for every sum ]T)(P).

Since th is cannot be smaller than the least upper bound, we have

V f(a ,b )< V f (a,c) + Vf (c,b).

The asserted equality now follows from the two inequalities. □

Theorem 1.1.7. Let f be a real-valued, function defined on [a, b] and let

f be o f bounded variation on [a, b]. Let V be defined on [a, b] as follows:

V (x) = Vf(a, x) i f a < x < b , V(a) = 0. Then:

(i) V is an increasing function on [a,b].

(ii) V — f is an increasing function on [a,b].

Proof. For a < x < -y < b, w rite Vf (a, y) = Vf (a, x) + Vf(x, y). Then

V(y) - V(x) = Vf (a,y) - Vf (a,x) = Vf (x,y) > 0.

Hence (i) holds.

Consider D (x) = V(x) — f { x ) i f x £ [a, 6]. For a < x < y 

0 and (ii) holds. □


T h e o rem 1.1.8. Let f be a real-valued function defined on [a, 6]. Then

f can be expressed as the differencef is o f bounded variation on [a, b] 4=>

o f two increasing junctions.

Proof. (= > ) Suppose / is of bounded variation on [a, 6]. W rite f = V — D

where D is the function from the proof of Theorem 1.1.7. Now V and D —

V — f are increasing by Theorem 1.1.7.

(4 = ) Suppose f = g — h where g, h are increasing. Then g, h are mono

tonic and so are o f bounded variation by Theorem 1.1.1. Now g — h is o f

bounded variation by Theorem 1.1.4, and so / is of bounded variation. □

This representation is not unique! I f / = f i — fa is one representation of

/ as the difference o f two increasing functions, then f = ( f i +g ) — (/2 + g),

where g is an a rb itra ry increasing function, is another representation o f / .

I f g is strictly increasing, so are f i + g and / 2 + g, therefore Theorem 1.1.8

holds if “ increasing” is replaced by “s tric tly increasing” .

T heo rem 1.1.9. Let f be a real-valued function defined on [a,b] and let f

be of bounded variation on [a, 6]. I f x G (a, 6], let V (x ) = Vf(a,x) and put

V(a) = 0. Then every point o f continuity o f f is also a point o f continuity

o f V . The converse also holds.

Proof. (4 = ) V is monotonic as i t is increasing, hence the righ t- and left-hand

lim its V ( x + ) and V ( x - ) exist fo r each point x in (a, b). By Theorem 1.1.8,

the same is true of f ( x + ) and f ( x - ) .


I f a < x < y < b, then 0 < |f ( y ) — f ( x ) \ < V( y ) — V(x) by the definition

o f Vf(x,y) . Letting y - * x, we see 0 < |f ( x + ) - f ( x ) \ < V{x-\-) — V(x) .

S im ilarly, 0 < | f ( x ) — f ( x —) \ < V(x) — V ( x —). These inequalities im ply

th a t a po in t o f continu ity of V is also a point o f continu ity of / .

(= ^ ) Let / be continuous at the point c in (a, b). Given e > 0, there

exists S > 0 such that 0 < \x — c\ < 5 implies 0 < \ f (x) — f (c) \ < e /2. For the

same e, there exists a partition P of [c, 6], say P = {xo, x i , . . . , xn}, x0 = c,

xn — b such that

fc=l

Adding more points to the pa rtition can only increase the sum Yh |A/fc|> so

we may assume 0 < x\ — xq < 8. This means tha t

IA/i| = | /(xi) - / ( c ) | <

As { x i ,X 2 , . . . , xn} is a p a rtition of [xi,b\, we have

V>(c,») - | < | + E lA A I < | + V f a , b).k=2

Hence

Vf (c,b) - V f {xu b) < e.

B ut 0 < V{x \ ) — V(c) = Vf ( a ,x i ) - Vf(a,c) = Vf (c ,x i) = Vf(c,b) —

Vf(x i ,b ) < e. Hence 0 < x x — c < 6 implies 0 < V '(x i) — V(c) < e. This

shows V (c+ ) = V(c). S im ilarly, V (c—) — V(c). □

T heorem 1.1.10. Let f be continuous on [a, 6]. Then

f is o f bounded variation / can be expressed as the difference

on [a, b] o f two increasing continuous Junctions.

Proof. W rite / = V — D as in Theorem 1.1.7. Then V and D are increasing

by Theorem 1.1.7, and are continuous by Theorem 1.1.9. □

As in Theorem 1.1.8, Theorem 1.1.10 holds i f “ increasing” is replaced by

“s tric tly increasing” .

Theorem 1.1.11.

/ is o f bounded variation f b= > the Riemann integral / f ( x ) dx exists,

on [a, b] Ja

Theorem 1.1.11 is proved in [4, pp. 207-212].

The converse of Theorem 1.1.11 does not hold, th a t is, a function may be

Riemann integrable w ithout being o f bounded variation. Consider

/ ( * ) = ( ° < :C- 1’[ 0, x = 0.

Then / is defined on [0,1]. As

J r f ( x ) dx = £ dx = [2x ll2}\ = 2 - 2e1/2,

we have f { x ) dx = 2 so the Riemann integral exists. Consider the pa rti

tion P o f [0, 1] w ith P = { 0, J , J, J , . . . , a } . Then A / i = f ( x i ) - f ( x Q) = n i

Suppose f is o f bounded variation on [0,1]. Then there exists a positive num

ber M such th a tn

n 1/2 = IAAI Jt=i

but th is fails whenever n > M 2. Hence / is not of bounded variation on

[0, 1].

1.2 John’s theorem

F irst we give John’s theorem, which was stated and proved by John in 1934

in [20],

T heo rem 1.2.1. Let f be a real-valued function on (—oo,+oo) which is

periodic o f period 1 and of bounded variation on [0,1]. I f c > 1 is a given

rational number, then

£ ^ f ( S ) = lo g c i m i v ' (1 2 1 )

where an(c) is defined as follows: set c = p/q, where p and q are coprime

integers with q > 0, then

an(c) = <

0, i f P in , q fn ,

-p , i f p \ n , q fn ,

q, i f p f n , q \ n ,

q - p , i f p \ n , q \ n.

Our form ulation is sligh tly different from John’s original statement which

is as follows:

T heo rem 1.2.2. Let g be a real-valued function on (—oo, +oo) which is

periodic o f period 1 and of bounded variation on [0,1]. I f c > 1 is a given

rational number, then

£ { * ■ - i | i ) = logci 9{y) dy ' (1-2-3>fo r the same definition o /a n(c) and fo r any t G R.

We show th a t the two versions of the theorem are equivalent.

Proof. (Theorem 1.2.1 = > Theorem 1.2.2) Let g be a real-valued function

on (—oo, +oo) which is periodic of period 1 and of bounded variation on

[0,1]. Let c > 1 be a given rational number, and let £ G l .

Set f ( x ) = g(t — x ). Then / is a real-valued function of (—oo, oo). Note

th a t / is periodic of period 1 as f ( x + 1) = g(t — (x + 1)) = g(t — x — 1) =

g ( t - x ) = f (x ) .

Let {xo = 0, x i , . . . , x n = 1} be a pa rtition of [0,1].

I f t e Z we set = 1—xn_fc, k = 0 ,1 , . . . , n, so th a t {xq = 0,x'1, . . . , x ,n =

1} is a pa rtition of [0,1]. Since g is of bounded variation on [0 ,1], there exists

M > 0 such th a t

~ 9 {x 'k- 1 ) | < M ./c = l

Hencen

^ | p ( l - xn—k) - t f ( l - x n_fc+ 1 ) | < M .fc=i

As g is periodic w ith period 1 and t G Z , we have71

5 3 19 {t - xn- k) - g(t - xn_fc+i)| < M.k= 1

Next, as f ( x ) = g(t — x), we deduce tha t

n

^ 2 \ f ( Xn-k) ~ f(Xn-k+l) I < M, k=l

th a t isn

5 3 l / f a n —fc + l ) “ f ( x n - k ) I < M. k= 1

Changing the summation variable from k to I = n — fc + l , we obtainn

1=1

so tha t / is of bounded variation on [0, 1].

I f t ^ Z, then

Xq = 0 < t - [t] < 1 = xn.

Let m be the unique integer € {0 ,1 , . . . , n — 1} such tha t

t M ^ 3'm+l-

Set

x'Q = 0,

x [ = t - [i] + 1 - xn,

®n—m t — [t] + 1 — X m+ i ,

* n—m+1

Then {x'0, x'x, . . . , x'n_mJrl} is a pa rtition of [0,1]. Since g is o f bounded vari

ation on [0, 1], there exists M \ > 0 such th a t

n—m + l

X ) Is K ) -0 (a 4 - i) l ^ M i-k= i

Hencen—m

1 “ M + 1 “ x n + l - k ) - g ( t - [t] + 1 - Zn+2-fc)|k=2

n—m

fc=2

As f ( x ) = g(t — x ) we deduce tha t

n —m

53 l / ( M - 1 + Z n + l- fc ) - f ( [ t ] - 1 + X n +2-fc )| < M i .

k=2

As / is periodic of period 1, we obtain

n —m

53 l / ( X n + l- i fc ) ~ / ( z « + 2 - f c ) l < M i .

k= 2

Changing the summation variable from k to I = n — k + 1 , we obtain

n —1

53 \ f ( x i ) - f ( x i + i ) \ < M i,l=m+1

tha t is,

equivalently

n —1

53 i/c^+i) - /(^)i < Mi,l=m+1

5 3 — /(^ fe - i) i < M i.k = m + 2

As <? is of bounded variation on [0,1], by Theorem 1.1.3 g is bounded on [0,1].

As g is periodic o f period 1, g is bounded on (—oo, oo). As f ( x ) = g(t — x),

f is bounded on (—oo, oo), say |/(x ) | < K for a ll i £ l Hence

n n

\ f ( x k ) ~ f ( X k - l ) \ = \ f ( x m+1) ~ f ( x m) \ + \ f ( x k) - f ( x k- i ) \k=m + 1 k = m + 2

< | / ( Z m + l ) | + l / ( * m ) | + M i

< 2 K + M i.

S im ilarly by form ing a pa rtition from Xq, Xi , . . . , xm we can bound J2k=i I f ( x k)-

f ( x k — 1)|, proving th a t / is o f bounded variation on [0, 1].

Lastly using Theorem 1.2.1, we have

y an(c) / logn \ _ y an(c) , f \ogn^ n 9 \ lo g c ) ^ n V logC>

: f f ( y )Jo

logo I f ( y ) dy )

log c g(t — y) dy Jo

r t - lr t - l= —logc g(z) dz (w ith z = t — y)

= log c g(z) dz Jt-1

= log c g(y) dy.Jo

(Theorem 1.2.2 =4> Theorem 1.2.1) Let / be a real valued function on

(—00, 00) which is periodic o f period 1 and of bounded variation on [0, 1].

Let c > 1 be a given rational number.

Set g(x) — f ( —x). Then g is a real valued function on (—00, 00). Note

tha t g is periodic of period 1 as g(x + 1) = f ( —(x + 1)) = f ( —x — 1) =

f ( - x ) = g(x).

Let xo = 0, x i , . . . , xn = 1 be a p a rtition o f [0,1]. Then

x'Q = 1 - Xn, x'x = l — rrn_ i , . . . , x'n = 1 - x0

is also a pa rtition o f [0, 1].

Since / is of bounded variation on [0,1], there exists M > 0 such th a t

X ) If ( x 'k) - / (Zfc-i) l < M and J 2 1/(1 - x'n_k) - / ( I - < _ fc+1)| < M.fc = l k = l

As / ( I - x'n_k) = f ( - x n- k) = g(xn- k), we have

n

23 \9(Xn-k) ~ g(Xn-k+1)| < M.fc=1

Setting j = n — k + 1 gives 2 3 \g(xj~i) - g(xj)\ < M , and, as we mayj = l

change the order due to the absolute value,

71

- 9 i xj - i ) \ < M >3=1

so g is o f bounded variation on [0, 1].

Lastly, setting t — 0 in Theorem 1.2.2, we have

an{c) f l o g n \ _ ^ an(c) f log n \E u n W £ I iV & n \ _ V - a n \ u) t , _

n 7 Vlog c ) ~ ^ n 9 \71=1 \ o / n _ j \

= log c [ g(y) dyJo

= log c I f ( —y) dy I of f i - v )Jo

= log c f ( l - y ) d y Jo

= —logc f ( z ) dz (w ith z = l — y)Jo

= log c [ f ( z ) dz,Jo

as required. □

CHAPTER 1. JOHN’S THEOREM

1.3 Proof of John’s theorem

23

Let c > 1 be a rational number. Set c = p/q, where p G Z , q € N and

gcdfp, q) = 1. Set A = — . Let a e E b e such th a t 0 < a < 1. Let M e N.logc

We begin by lis ting a few simple properties o f c,p, q, A, a and M th a t we

w ill need in the proof of John’s theorem.

Lem m a 1.3.1. (i) p > q,

(ii) log c > 0,

( iii) A, A-1 > 0,

(iv) a + A log M q < A log Mp,

(v) {A lo g M p } = {A log M g },

(vi) . logq = lQgP 1logP — log? log p log q

Proof, (i) As c = p/q > 1 and q > 0 we have p > q.

(ii) As c > 1 we have log c > log 1 = 0.

(iii) By (ii) we have

1 A 1 1> 0, — = log c > 0.logc A

(iv) We have

a < 1 = » a < AA-1 =>• a < A log c

=$> a: < A log - ==> a < A log” & q ~ ° M q

==> a < A (log M p — log Mq)

= » a + A log M q < A log Mp.

(v) Also

{A log M p } — { A log M p — A log M q + A log M q }

= {A log ^ + A log M q }

= {A • A-1 + A log M q ]

= {1 + A log M g}

= {A log M g }.

(vi) F ina lly

logg — (log p — log g) + log p logp= —1 +

logp — log q logp — log q log p — log q

Next we recall Euler’s constant

7 = lim -j — log = 0.5772156649__

We w ill need the follow ing estimate.

Lem m a 1.3.2. Let x ,y € R be such that x > y > 0. Then

£ I = l o g ( * ) + < > ( ! ) .£ £ m \ y j \ y j

y < m < x

as y +oo, where the constant implied by the O-symbol is absolute.

□

Proof. The asym ptotic formula

T - = logy + 7 + O ( - ) , y -> oo ^ m V y Ime N m < y

is well-known, see for example [3, p. 55]. The im plied constant is absolute.

Thus for x > y > 0 we have

1E 1 = E 1-Em " , to " , tom€N meN meN

y < m < x m < x y < m

= ^ lo g x + 7 + 0 - ^ logy + 7 + O Q

= iosG)+oS)’as y —» oo. □

We now make use of Lemmas 1.3.1 and 1.3.2 to prove the follow ing result.

Lem m a 1.3.3.

lim Y — = ctrA-1 .M -* o o TOmeN

M q < m < M p 0 < {—A log m }<a

Proof. We have

E 1 = E i" , TO " , TOm€N m€N

M q < m < M p M q < m < M p0 < {—A logm }<a 0 < —A log m —[—A log m ]<a

E -1TO^ez meNE

A log M q < £ < \ log JVf p - f l M q < m < M p0 < —A lo g m —[—A lo g m ]< a

[— A lo g m ] = —I

Elez

A log M q < t< —\ log M p+1

E€eN

A log Mq<£<A log M p+1

E£ez

A log Mq<£< A log M p+1

E 1m£N

M q<m <M p 0 < —A log m + i< a

[—Alogm]= —£

E -1me N

M q<m <M p 0 < —A log m+£<a

me N M q<m <M p e-a t

e A < m < e *

We now break the sum over £ into 4 subsums S i, S3, S4 according to the£ _ Q ^

sizes o f e“ and e* relative to M p and Mq. We have

E E - = Si+S2 + S3 + S4,£eZ meN

A log M q < t< \ log M p+1 M q<m <M pI —C*

where

e a <m<e>

= E E s-iez meNA log M q < £< \ log M p+1 M q<m <M p

e-a e a < M g

M p <e^

= E E s-£ez meN

A log M q < £< \ log M p+1 fcr.aI — Of

Mq<e A

Mp<e^l—Ot

e A < M p

* - E E s-£ez meN

A log M q <£< \ log M p+1 M ?<rn< efe~TT<Mq

e% <M p

e A < m < M p

s<= E £ b£ez me nA log M q <£< \ log M p +1 1

£ q C A <771^6 AMq<e A

< M p

We note th a t S2 = 0 unless e1 2 < M p. Also in the sum S3 we observe tha t

M g < e* as A log M g < A

F irst we examine S i. By Lemma 1.3.2 we have

* - £ M 9 * ° ( i ) )A log M q<£<A log M p+1 A log M p<£<a+A log Mq

= 0,

as a + A log M q < A log M p by Lemma 1.3.1 (iv).

Next we determine S2. By Lemma 1.3.2 we have

* = E M ¥ s ) + ° ( - k£€Z

A log M q < £< \ log M p+1 c*+A log Mq<£<a+A log Mp

A log Mp<£

, _ , <-ae a / \ e a .

- a J£eZ ' \ o A /

A log Mp<£< A log M p+as M ; ? M +

by Lemma 1.3.1 (iv). We consider 3 cases according as {A log M p } = 0,

0 < {A log M p } < 1 — a, o r l — a < {A log M p } < 1.

I f {A log M p } = 0 then A log M p G Z so i = A log M p. Thus

& = log + O ^ 1g(A log M p —a )/A J \ g(A log M p —a)/A

- + o ( — \. a v m :

I f 0 < {A log M p } < 1—a then [A log M p \+ 1 > A log M p + a so A log M p £

Z and no such integer £ exists so S2 = 0.

I f 1 — a < {A log M p } < 1 then A log M p Z and [A log Mp] + 1 <

A log M p + a and £ = [A log Mp] + 1. Then

M p \ ^ / 1S2 log

log

g([A log M p]+1—a)/A

M p

+ 0

g(A log M p —{A log M p }+ 1 —a)/A

1+1A log Mp) ^ / 1 \

^ M J

e ([A log M p]+1—a)/A

+ 0 g(Alog Mp— {A log M p }+ l-c t) /A

Hence

log (e a " a“ J + 0

& 1 + {A log M p } ( J _ 'A + V M .

i +0(s)-S2 = 0,

a - l + {A lo g M p } ( 1 ' A + ° [ m .

{A log M p } = 0,

0 < {A lo g M p } < l —a;,

1 — a < {A log M p } ,

Now we tu rn to the evaluation of S3. We have by Lemma 1.3.2

1S3 = E

tezA log Mq<£<A log M p+1

£<o+Alog M q £ < \ log M p

h ^ W q ) + 0 K M

Eeez

A log M q<£<a+A log Mqlog|^ ) +0( s /

by Lemma 1.3.1 (iv).

Here we consider 3 cases according as {A log M q } = 0, 0 < {A log M q } <

1 — a, or 1 — a < {A log M q } < 1.

I f {A log M q } = 0 then A log M q € Z and we have

I f 0 < {A log M q } < 1 - a then [A log Mq] + 1 > a + A log M q so

A log M q 0 Z and no such integer £ exists so S3 = 0.

I f 1 — a < {A log M q } < 1 then [A log M g] + 1 < A log M q + a and we

have

S3 = log[A log Mql-fl \

e a

M q

[A log Mq] + 1H i

log M q + O l —

1 — {A lo g M g } + 0 / J _ AA M .

Hence

Ss =

0 { i ) ,

0,1 — {A log M q }

A

which is equivalent to

o ( A ) ,s3 = < 0 ,

1 — {A log M q }

{A log M g } = 0,

0 < {A log M g } < 1 — a,

+ 1 - » < {A lo g M g },

{A log M p } = 0,

0 < (A log M p } < 1 — a,

+ ° [ j j r ) > 1 - a < {A lo g M p },

as (A log M p } = {A log M g }, by Lemma 1.3.1 (v).

Lastly we consider S4 . By Lemma 1.3.2 we have

A log M q<£<A log Mp+1 a+A log Mq<£< A log Mp

a+A log Mq<£<A log Mp

We consider 3 cases according as {A log M p } = 0, 0 < {A log M p } < 1—a,

or 1 — a < {A log M p } < 1.

I f {A log M p } = 0 then A log M p e Z and no such integer I exists so

SA = 0.

I f 0 < {A log M p } < 1 — a then I = [A log Mp] and

I f 1 — a < {A log M p } < 1 then no such integer £ exists and S± = 0.

Hence

P utting i t a ll together, we now look at the required sum S\ + S2 + S3 + 64

in 3 cases according as {A log M p } = 0, 0 < { A log M p } < 1 — a, o r l — a <

{A log M p } < 1.


0 {A log M p } — 0,

0 < { A log M p } < 1 — a,

1 — a < {A log M p } < 1.

I f {A log M p } — 0 then

S l = 0 , f t = 2 + 0 ( i ) . S 3 = o ( i ) , S 4 = 0 ,

so

S± + S2 + £3 + S4 = — + 0 •

I f 0 < { A log M p } < 1 — a then

Si = 0, S2 = 0, £3 = 0, SA = j + o ( j j ) ,

so

Si + Si + Ss + S ^ j + O ^ y

I f 1 — a < { A log M p } < 1 then

S1 = 0, ga = a ~ 1 + ( l0gM p}+ ° Q _ ) .

S4 = 0, g3 = l - { A b g M p } + 0 Q _ ^

SO

Sl + S , + 8, + 3t = ^ - l + {AI°gMp} + l - { A l o g M r f + 0 / lA V 1VI

= x+0(f )-Hence in a ll three cases we have

s1 + s2 + s3 + s4 = j + o (J ^ J

so

lim V ' — =M-* 00 <' m AmCN

Mq<m<Mp 0<{—Alog m}<a

as required.

CHAPTER1. JOHN’S THEOREM 32

For a € [0,1] we define a special step function <j)a as follows: for y G R

we set

M y ) = 0) y e R,

0 i ( i ) = 1) y € M,

and for a ^ 0,1

M v) = { x ’ 0 “ W < (1-3.1)[ o, a < { y } < 1.

Lemma 1.3.4.

lim V "' — (f>a(—A logm ) = aA-1 .M -* oo 771

m€N M q<m<M p

Proof. As </>o(y) = 0 for a ll y, Lemma 1.3.4 is tr iv ia lly true for a = 0 and we

may assume th a t 0 < a < 1.

By the defin ition o f <pa we have

£ ^ f e ( - A l ° g m ) = £ 1m 6N meN

M q<m<M p M q<m<M p0 < { — A logm }< a

and the asserted result follows by applying Lemma 1.3.3. □

Lemma 1.3.5. Let f be a periodic step function o f period 1. Then

d im — / ( - A l o g m ) = A-1 / f ( y )d y .

M q<m <M p

Proof. As / is a periodic step function of period 1, / can be b u ilt up as a

linear combination o f a fin ite number of the step functions 4>a(y) w ith differ

ent parameters a. Hence the asserted result holds by repeated application of

Lemma 1.3.4. □

Lemma 1.3.6. Let f be a periodic function of period 1, which is Riemann

integrable on [0,1], Then

lim — /( -A lo g m ) = A-1 f f ( y ) dy.M~°° m J 0

M q<m <M p

Proof. Let e > 0. As f ( y ) is a Riemann integrable function on [0,1] and is

periodic of period 1, there exist two step-functions and <£(?/) o f period

1 such tha t

m < f ( v ) < H v ) (1-3.2)

and

[ (<% ) - 4>{v)) dy < e, (1.3.3)Jo

see for example [5, Sections 1.1, 1.2, pp. 11-26]. Set

Sm {4>) = 53 - 0 ( -A log to), to

me N M q<m <M p

Sm ($) = 53 —$(~A logm).mmeNM q<m <M p

Then, by Lemma 1.3.5, we have

lim SM{<f>) = A-1 [ 4>{y) dy,M —> oo J Q

lim SM($) = A-1 [ $(y) dy, (1.3.4)M —*oo J o

and from (1.3.2) we have

Sm {4>) < 53 - / ( - A l o g m ) <m me N M q<m <M p

Prom th is and (1.3.4) we conclude

/*i______________________ i1 4>{y) dy < lim in f V " — / ( - A l o g m )

J o Z a mM q<m <M p

< l im sup ^ 2m€N m

M q<m <M pM q<m <M p

JO

and applying (1.3.3) gives the asserted result. □

We now have the results needed to prove John’s theorem (Theorem 1.2.2).

Proof. We follow the proof of John’s theorem given by Rademacher in [27,

pp. 170-173], but expanding on the details where necessary. We note th a t

the proof only requires g to be Riemann integrable (recall from Theorem

1.1.11 th a t functions of bounded variation are always Riemann integrable).

We wish to study the expression

for a Riemann integrable function g and an(c) as defined in (1.2.2). We begin

by showing tha t we need only consider such N which are divisib le by pq.

For 0 < R < pq, as an(c) and g are bounded, we have

i™ E ir9(i_Alogn)n = 1

Npq+R Npq+R

g(t — X log n) < A ^n=Npq+ln=Npq+ 1

fo r some constant A.

By Lemma 1.3.2 we have

Npq+R 1

n=Npq+ 1

= log

1 nNpq<n<Npq+R

Npq + R N+0(vNpq

= log{ I + + 0 ( j j

Hence

proving our claim.

Now set

N p q )

lim Y — = log 1 = 0,oo *n.

Npq+R

N -* oo e~~J n n=Npq+ 1

Mpq / \

-5m (5) = E “ A logn).1n = l

Then, using the definition of an(c), we obtain

Sm (9) = E A l o g n ) - E - 0( t - A l o g n )* * r?. •> ■ * r i

l<n<M pqq|n

Mp

\ < n < M p qp|n

Mq" y 2 2= y \ - g ( t - A logm ?) - V - <?(* - A logm p).

i TO ' TOm = l m = l

Using Lemma 1.3.1 (v i), we have

log to + log qA log mq =

which gives

? (i - A logm ?) = g ( t ■

as g is periodic o f period one. As

g ( t - A logm p) = ? ( t -

log TO+

logp - 1,logp — log q logp — log q logp — log q

log to logplog P - log q logp - log q

log to logp \logp - lo g q logp log q ) ’


we deduce tha t

log p log q log p log qlog m log p

We set t = A logp, for i f (1.2.3) is true for any special value to, i t is also

1.4 Evaluation of certain infinite series using

John’s theorem

We begin by choosing f ( x ) = 1 (x £ R). C learly / satisfies the conditions of

John’s theorem.

T heo rem 1.4.1. Let p and q be coprime integers w ithp > q > 0. Then

true for any other t, as g{x) and g(x — to + t) regarded as functions of x are

both periodic and Riemann integrable.

Hence a ll we need to show is

Mp .

lim SM(g) = lim V — s (-A lo g m ) = A' 1 / g(y) dy

B ut th is was shown in Lemma 1.3.6, and the proof is complete. □

where an is defined in (1.2.2).



Taking p = 2 and q = 1, we obtain the well known result

1 - - + = log 2.2 3 4 &

W ith p = 3 and q = 1 we have

, 1 2 1 1 2 1 1 2 , 01 H------------ 1------1------------ j------1------------ 1------ = log3.

2 3 4 5 6 7 8 9 6

W ith p = 3 and q = 2 we obtain

2 3 2 1 2 3 2 1 , 3---------- 1------------ 1------------ 1----------------(- .. . = Jog —.2 3 4 6 8 9 1 0 12 6 2

These series are a ll o f the form

00E Qn

n ’7 1= 1

where {a n} is a repeating sequence of integers. Such a series is called a

harmonic-type series. The harmonic series itse lf demonstrates tha t not a ll

harmonic-type series converge. Lesko [23] has recently established a necessary

and sufficient condition for a harmonic-type series to converge, namely

oo

A harmonic-type series — with repeating coefficients a i,a 2 , . . . , a ki n

n = lk

converges i f and only i f ^ a* = 0.i = 1

Series o f the formoo

E O-n k r - ’

n = l

where { a „ } is a repeating sequence o f integers and A: is a given real number,

have been treated by Longuet-Higgins [25].



For an integer b > 1 i t is convenient to define

f b - 1, i f b | n, en = €n( b ) = { (1.4.1)

[ - 1, i f bj(n,

as in [8, p. 22] and [12, p. 261]. C learly

an(b) = -e n(&), en(2) = ( -1 ) ” . (1.4.2)

Taking p = b and q = 1 in Theorem 1.4.1 we obtain

C o ro lla ry 1.4.1. Let b be an integer > 1. Then

y ; ^ = - io g 6.r)

71=1

Corollary 1.4.1 is well-known. I t appears for example in [16, Problem 31],

[19], [22, p. 136].

We next use John’s theorem to evaluate the in fin ite series

^ (-1 )"-* flogn \ kn \ log 2 /

for any k E N. Recall th a t {y } = y — [y] is the fractional pa rt of the real

number y.

T heo rem 1.4.2. For k G N , we have

( ~ l ) n f logn V ___ 1_log 2.

Proof. In John’s theorem we take c = p/q = 2/1 and f { x ) = {a ;}fc (a; G

Then / is real valued on (—00, 00) and is periodic of period 1.



By (1.4.2) we have

a „(2) = —e„(2) = - ( - 1)» = ( - 1) " - 1.

Set

g(x) = xk, x e [0, 1],

, f s f 0, X e [0,1),h(x) = I

[ 1, s = 1.

Then g is o f bounded variation on [0,1] as i t is monotonic on [0,1] and h is

obviously of bounded variation. Now / = g — h so / is o f bounded variation

on [0,1] by Theorem 1.1.4.

Then, by John’s theorem, we have

= log 2 [ yk dy Jor yk+l 1 1

= log 2 1 y_k + l

1 log 2,k + 1

and the asserted result follows. □

Taking k ~ 1 in Theorem 1.4.2 we obtain

C o ro lla ry 1.4.2.

f^ (- l)" I logn 1 l , , . n§ n \ log 2 / 2 g '


Next we use John’s theorem to evaluate an in fin ite series which includes

° ° / 1 \7 1 r i _______ ^ kf s ( - l ) " f l o g n y n \ log 2 J

n = l

as the special case 6 = 2.

T h eo rem 1.4.3. Let k e N . Let b € N 6e such that b > 1. Then

OOy e J V r i o g r n i = _ _ l _ l o J

n 1 log 6 J Jfe + l &71=1 v '

Proof. In John’s theorem we take c = b. By (1.4.2), we see th a t

We choose f ( x ) = { x } k (x 6 R) so th a t / is real-valued on (—00, 00), is

periodic of period 1, and of bounded variation on [0, 1].

Then, by John’s theorem, we have

E T I 10* 1*71= 1 v

and the asserted result follows. □

°° € (b) f lo (ti/Qj] 1 ^F ina lly we use John’s theorem to evaluate the series > — ■ < —. , > .

£1 n \ log 6 /

T heorem 1.4.4. Let k € N. Lef a and b be integers with a > 1 and b > 1.

Then kr i o g ( n / q ) \ _ 1

^ n \ log 6 J fc + 1


Proof. We use John’s theorem in the form of Theorem 1.2.2. We choose

g(x) = { - x } k, c = b, * =

Then

®n(c) = €n(fi)

and John’s theorem gives

th a t is

^ n \ lo g t J k + 1 g ’

as asserted. □

1.5 The generalized Euler constants

Euler’s constant 7 is defined by

7 = lim [ V 1 - logn | = 0.5772156649.... (1.5.1)\ U 3 )

I t is well known th a t

^ i = log x + 7 + 0 ^ , (1.5.2)

as x —> +oo, see for example [3, p. 55].

The generalized Euler constants 7* (k = 0 ,1 ,2 , . . . ) are defined by

* = 0.1.2........ (1.5.3)n_>0° \ 3 k + 1 J


see [12, p. 259], where we understand log0 1 = 1. C learly

7o = 7* (1.5.4)

From [3, p. 70] we know th a t

£ ^ = i l 0 g’ * + 71 + 0 ( ^ ) , ( 1.5.5)

as x —> +oo.

n 2 V xn<x v

We next use the asymptotic formulas (1.5.2) and (1.5.5) to evaluate the00 jo jj

series ^ ( —l )n_1------- , which w ill be needed in the next section.n n = 1

Theorem 1.5.1.

E ( - l ) " - 1^ = —7 log 2 + log2 2., 71 71= 1

Proof. Let N € N. Then

2 N . 2 N , 2 N

= £ « - i7 1 = 1 7 1 = 1 71=1

IVo log 2n

^ 2n

log 2n

71=1

N

= -Ei n71=1

N

E log 2 + logn

i n71=1

logn- k « » E " E -n n

7 1= 1 71=1


Hence

Y ( - = V'!2«!!_log2y' I_y' ‘2«Iin -4-f n n n

n = 1 n= l 7i=l n=l

= 5 log2(2JV) + T1 + o ( ^ H )

— log 2 ( lo g N + 'y + O

- Q lo g > W + 11 + 0 ( ^ ) )

= Q (log 2 + log iV )2 - log 2 log N - i log2 N^j

+ (7 i “ 7 log 2 - 7 i) + 0

= Q log2 2 + log 2 log N + ^ log2 N - log 2 log iV - ^ log2 iV

-7 lo g 2 + o ( ^ )

= - log2 2 — 7 log 2 + O •

Letting N —> +oo we obtain

V " '(—l ) n -1 logn 1 2o---------------------- —7 log 2 + - log 2,

n 271=1

which gives the asserted result. □

Theorem 1.5.1 can be found in [12, p. 263] and [15, p. 288].

Our next result provides a generalization of Theorem 1.5.1.

T he o rem 1.5.2. Let b > 1 be an integer. Then

£ £n(5) = 7 log b - | log2 b.71= 1


Proof. For x > 1 we have

logn -<r-v m -,\l°g n v™' lognE « - w T = E f e w + D ^ - E - nn < x n<x n< x

- f t V ' lo g n _ y ' lo gnn n

n<x n<»xb]n

_ b y - log (fan) logn, bn " n

n<x/b n<x

= log6 Y ) i + E ! ^ - V l 2 L n“ n ' n “ n

n<«/fe n<x/b n<x

= lo g i, ( lo g f + 7 + o ( ^ ) )

- Q log2 x + 71+ )"logs

s= log b log s — log2 6 + 7 log b

~ log 6(2 log x - log b) + O

= “ log2 6 + 7 log 6 + O •

Letting x —> +oo we obtain

£ e” (6) ~ ~ = 7 log 6 ~ ^ log2 b.n= 1

□

Theorem 1.5.1 is the special case b = 2 of Theorem 1.5.2.

In 1972 Liang and Todd [24] proved the follow ing extension of Theorem

CHAPTER1. JOHN’S THEOREM

n = l t>=0 ' '

We generalize Liang and Todd’s result as follows.

T heo rem 1.5.3. Let b > 1 be an integer. Let k e N. Then

^ /1Alog* n log^+ 'JE ‘ • w — = E L I (log6) “ t + T '7 1 = 1 V = 0 V '

Proof. Let N 6 N. Then

°° i fc , 1.en(6 )__ = hm 2^en(6)——

n at—»oo ' n71= 1 71= 1

{ JW 1 fc 1 «V ( 6 - 1 ) ! ^ + E ( - I ) ^ n

' n y nn = l n = l6|n tyVi

{ Mb 1 k Mb , ky » ^ - y y - D — ”

' n *—• nn = 1 n = lb|T.

{ IV 1 fc L 1 fc 'y - r log nb log n 2 - i n 2 ^ n7 1= 1 71= 1

— (logn + lo g 6) fc log* n

fc,log n

= nm yN-+00 n n

1 7 1= 1 71= 1

{ N 1 k / u \ Nb

7 1= 1 S=0 ' ' 71= 1

{ k / U \ N 1 « Nb 1 ft

S Q ^ E ^ - E ^ "5=0 x ' n=l n=l



log k n log k+1N b \ logfc+1 Nb

s—0

Nb

E n k + l / f c + lkTl=l

k fk= E L lo^ sb j s - i k

s= 0 ' S

N->oo \ E-/ \ s I S + l f c + l

■ 5 C R " ’-

k - i /jfcN

,• S 1 /, t , ,» M log^ 6 log^+ JV6'+ i t e S o ( r + r ( g g ^ “ T + i r + i - ,

- E(TW^»/lo g fc+1 Nb log*+1 b logk+l N b \

+ limN-* oo \ A; + 1 A; + 1 k + l

f k \ k_ log* * 16= E ( J l0 b ' i ' - ~ k + T -3 = 0

□

Theorem 1.5.3 can also be deduced from Corollary 1 and Proposition 4

in [12, pp. 260-261].


CHAPTER 1. JOHN'S THEOREM 47

Before proceeding we recall Euler’s summation formula.

T h eo rem 1.5.4. I f f has a continuous derivative f on the interval [y,x],

where 0 < y < x, then

= / f ( t) dt + [ ( * - [ * ] ) / ' ( * ) <**y<n<x V y

+ / ( * ) ( N - x) - f(y)(ly) - v)‘

Proof. See [3, p. 54]. □

T heorem 1.5.5. Let k E N . Then

E log k n log k+1n f\o g k x '- Z — = 7. , , + 7fc + O

n<xn k + 1

as x oo.

Proof. In Theorem 1.5.4 we choose y = 1 and f ( x ) = ^ - £. C learly

k logfc_1 x — logfc x/ '(* ) = X*

is continuous on [ l,x ]. Then Euler’s summation form ula gives

l< n < x

k\ogk l t — logfet 'dt

Now

/

log XX

(N - x).

log Ktdt =

log k+1t k + l

x _ logfc+1 X

i k + 1

' k log t — log t

= / V w > (

J dt

k log*-1 1 - logk t st 2

dt

poo

/ (t-W)J %h\ogk l t — \ogKt '

a : t 2.fc- i o r t

t

— + O^log fcx '

for x sufficiently large and a constant Ak\ and

! ^ ([a;] _ x) = o ^X

Thus

En<x

log* n log*+1xn

+ Ak + O'\ogk x '

As

k + l

log* n logfe+1 x n k + 1

we deduce th a t Ak = 7&, so tha t

log* n log*+1x

lim f v 'x—*oo I / \ n < x

En< x

n k + l + 7k + 0

x

— Ik i

^ lo g *s '

dt

as asserted. □


1.6 Dr. Vacca’s series for 7

Euler’s constant 7 is defined by means o f a lim it, namely

7 = lim f 1 + 7H 1- - - lo g n ) .rc—>oo y i n J

Vacca [33] in 1909 was the firs t person to show th a t 7 can be expressed by

means o f an in fin ite series, namely

( - 1)'

71= 1n

lognlog 2

The series (1.6.1) has become known as “D r. Vacca’s series for 7 ” . We

obtain (1.6.1) by deducing i t from Corollary 1.4.2 and Theorem 1.5.1. For

other proofs o f (1.6.1), see Addison [1], Bauer [6], Gerst [13], Koecher [21],

and Sandham [31].

Theorem 1.6.1.( - 1)”

n = ln

logn log 2

Proof. As [y} = y — {y } , we have

£71=1

( ~ l) rn

logn log 2 E ( - 1) " /lo g n J 'lo g n 'lN

^ n \ l ° g 2 \ lo g 2 j ;

_ y > ( - l ) n logn y s ( ~ l) n f lo g n ) n log 2 n 1 log 2 J

n = l ° n = l v y^_-| 7»

lo§ 2 S n l0gn 5 n \ log 2 /_ ^ ( - 1)^ f lo g n )

71= 1

log 27 log 2 - ^ log2 2 ) - log 2



by Theorem 1.5.1 and Corollary 1.4.2 respectively. Hence

E71= 1

( - i)"n

logn log 2

= 7 _ I l og2 + I lo g 2 = 7 ,

as asserted. □

As a consequence of Theorem 1.6.1 we have the follow ing alternative series

for 7 .

T heo rem 1.6.2.

2n+1 - 1

Proof. We have

y . / 1_ _ 1 _ 1 ,^ n \ 2 n 2n + 1 2n+1 - 1 n = l ' >

00 2n+1—1

E»En= 1 m=2n 00 2n+1- l

( - 1) 'm

= E En—1 m—2"

00 2n+1- l

( - 1) 'm

-n

n = 1 m =2n

00 -Qm

by Theorem 1.6.1. □

Our next result generalizes the Vacca series for 7 .



T heorem 1.6.3. Let b be an integer > 1. Then

00

7 = Een(6)

n = 1n

lognlog6

Proof. We have

E7 1= 1

Cn(6)n

logn log b

_ en(b) f log n f logn ) \ ^ n \ log 6 \ lo g fe /y

- OO 1 0 0

k E ^ - Elog 6

1log 6

7-

71= 1 71= 1

en(b) f log n n \ n ,

7 log 6 — i log2 6^ - ^ log b

□

Theorem 1.6.3 is due to Berndt and Bowman [8, p. 22, Theorem 2.6].

Our proof is much simpler than th a t of Berndt and Bowman, which involves

complicated integrals.

Exactly as we proved Theorem 1.6.2, we can prove the follow ing result as

a consequence of Theorem 1.6.3.

T heo rem 1.6.4. Let b be an integer > 1. Then

7 = ? _ + ________’ v bn bn + 1 bn+1 - I J 'n = l x '

The follow ing generalization of Theorem 1.6.3 was proved recently by

Berndt and Bowman [8, Theorem 2.8, p. 23].



T h eorem 1.6.5. Let a and b be integers with a > 1 and b > 1. Then

t = Een(&)

nlog(n /a)

log b

a—1 1

- lo§ a-71= 1

Proof. We have

En~a

£n(b)n

lo g in /a ) ' log b

en(b) / log (n /a ) f log(n /a ) } \^ n V lo g b I log b ] )

— _ 1 _ V f h ) lo g a £n(b) v e»(6) f log 6 " n log 6 " n n 1

° 7 1= 1 ° 7 1= 1 7 1= 1 V'

e»(6) f log(re/a) log 6

= ^ (^ log fe - | lo g 2 fc) - § | (- 'o s ^ - (- | I ° S &)

= 7 - ^ lo g 6 + loga + ^ log&

= 7 + log a.

Hence

7 - E= E

log (n /a) log b

— log a

log (n/a)log a.

71=1

Fina llya—1

E71—*1

(&)n

log (n /a ) log b

0—1

- E €n(b) + 1 log(n /a)

71= 1

a—1

n

£ -' nn —1b \n

lo g (n /a )’ log b ,

log 6a—1

a—1 ..

- E 1“ n71=1

lo g (n /a )' log b

U*—X -j

E 171=1

log (n /a )log 6


E 1\<n<a /b

log (bn/a) log 6

log bE H 1+\<n<a /b

E - - E -■4—', n ' n1 <n<a/b a/b<n<a

1 <n<a

log (n /a ) '

log (n /a) log b

E 1l< n < o

log(n /a) log 6

log (n /a )log b

E ; - E £(-1)~n<a/b

E -1 < n < a a —1 -

E v

, n ' n1 <n<a/b a/b<n<a

n= 1

□

Again our proof of Theorem 1.6.5 is simpler than th a t o f Berndt and

Bowman [8, Theorem 2.8]. Berndt and Bowman remark th a t th is theorem

is “apparently equivalent to ” a theorem o f Glaisher [14] w ithou t giving any

details.

In Theorem 1.5.3 we evaluated the in fin ite series

QO 1 kX y n( b ) - ^ , b (> 1 ) € Z , k e Kn = l

We now have enough inform ation to estimate the sum

6 ( > 1 ) € Z , k e N ,n<x n

fo r large x.

T heorem 1.6.6. Let b, k G N with b > 1. Then

5-0

as x —¥ oo.

Proof. Let b, k G N w ith b > 1, and rr G R. Then

m log *n V ^/1 , . V ' 1°gfenE £- ( 6) — = 2 ^ ( 1 + e* (6) )— - 2 . —n < s n<x n<x

— $ logfc n v logfc nn<* n < x6jn

_ logfc bn y-v logfc n“ n Z - j n

n<x/b n<x

_ (log6 + lo g n )fc y -v lo g fcnn n

n<x/b n<x

= E s e Q ^ + ^ - e ^71< x / 6 5 = 0 N 7

A:fcN\ i i - u r logSn v - ^ logfcn- „ "E „

«=0 ' / n<x/b n<x

( logfe+1 x/b ( logfc x1 7 + 7 k + 0 1 &

k + l \ xk

rn E (*+1)log‘~'6 log‘+1 x / h + ' E © logl5=0 x 7 5=0 x 7

logfc+1 x (\o g k xl k + 0

k + 1 \ x


- sb 7*


as asserted.

k +- ( lo g 6 + lo g x /b )k+l - — j— logfc+1 bi K + l

s = 0+E log*+1X

k + lJ k + O

'logk x 'X

- Es=0k

= E

logfe s 6 7S —

logfc-s ft 7s -s=0

k +

1k +

□


Chapter 2

Radem acher’s Theorem

2.1 N otation

The follow ing notation is used throughout th is chapter.

Let K be an algebraic number field of degree n over the rational number

field Q. The ring of integers of K is denoted by O k - The number o f roots

o f un ity in K is denoted by w {K ), the discrim inant of K by d (K ) and the

regulator o f K by R {K ). The number o f real fields among the conjugate

fields o f K is denoted by r and the number o f nonreal fields by 2s so th a t

n = r + 2s. The structure constant of the fie ld K is the quantity

2 '+ V R ( K )

w (K )^ \d (K Y \'

Two nonzero ideals A and B o f Ok are said to be equivalent, w ritte n A ~ B,

56


CHAPTER 2. RADEMACHER’S THEOREM 57

i f there exist 0) £ Ok and /3 (^ 0) £ Ok such tha t

< a > A —< (3> B. (2.1.2)

Clearly ~ is an equivalence relation on the set of nonzero ideals of Ok - The

equivalence class containing A is denoted by [A]. Since the quotient field of

Ok is K [2, Theorem 6.1.5, p. I l l ] , (2.1.2) is equivalent to the existence of

c(y^ 0) £ K such th a t

A = cB. (2.1.3)

The norm of an element c £ K is denoted by N(c) [2, p. 222] and the norm

of an ideal A o f Ok by N (A ) [2, p. 143]. The greatest common divisor of

A and B is denoted by (A, B ). The follow ing result w ill be im portant in the

extension of John’s theorem to algebraic number fields due to Rademacher

[27, Theorem, p. 173].

T heo rem 2.1.1. Let c 0) £ K . Then there exist unique nonzero ideals

A and B o f Ok such that

A = cB, (A, B ) = < 1 > .

Proof. As c 0) £ K there exist a { ^ 0) £ Ok and b £ N such th a t

c = a/b,

see for example [2, Theorem 4.2.6, p. 85]. Let P i, . . . ,P m be the set of

prime ideals which divide either < a > or (or both). Then there exist

nonnegative integers a i , . . . , om and nonnegative integers b\ , . . . , bm such tha t

< a > = P 1a i- - -P f lm, = P * 1 ■ • • P ^m.

Reorder P i, . . . , Pm so th a t a* > bi fo r i = 1 ,2 , . . . , k and a; < bi for i =

k + 1, . . . ,ra . Set

= p ai-6i . . . p ak—bk Q _ p fck+l- “fc+i . . . p bm—am1 fe Aj+1 wi

Then

■Pi01 • • • pmm = < & > = < be > = c = cPx61 • • • P ^m

soD ai-6l p Ofc—6fc - p 6fc+l-Ofc+l P b r r x - O m

1 ' " k k+l m >

tha t is,

A = cB.

Since the only prim e ideals d ivid ing A are P i , , Pk, the only prime ideals

d ivid ing B are Pk+i, . . . ,P m, and P i, . . . ,Pm are d istinct, i t follows tha t

(A ,B ) = < 1 > ,

which establishes the existence o f A and B.

Suppose A! and B ' are nonzero ideals o f Ok such tha t

A! = cB', (A', B ') = < 1 > .

Then

A 'B = (cB ')B = (<zB)B ' = A B '.

Hence

A I A 'B .

B ut {A, B ) = < 1 > , so

A \ A ' ,

say

A ' = A M

for some ideal M o f Ok - Then

A M B = A 'B = A B ',

so

B ' = B M .

Hence

< 1 > = (A1, B ') = {A M , B M ) = M {A , B ) = M < 1 > = M .

Thus

A! = A < 1 > = A, B ' - B < 1 > = B ,

which establishes the uniqueness of A and B. □

E xam p le 2.1.1. Let K = Q (> /=5). We choose c = € K . We

determine ideals A and B o f Ok = Z + Z \ /—5 such that

A = ( 1 + ) B, { A ,B ) = < 1 > .

We /iave

N (1 + a /T 5) = 6 = 2-3, iV(2) = 4 = 22.

The prime ideal factorizations o f < 2 > and < 3 > are

< 2 > = P 2, < 3 > = PXP2,

where

P = < 2,1 + V --5 > = < 2,1 - > ,

P i = < 3,1 + >/=5 > , P2 = < 3 , 1 - V C 5 > ,

see /o r example [2, p. 265]. Clearly P ^ P1; P ^ P2, P i ^ P2, N (P ) = 2

and JV(Pi) = iV(P2) = 3. Now

< l + v/= 5 > = < 1 + vc 3 > < 1 - V c 5,2,3,1 + >/Z5>

= < 6,2(1 + \ / —5), 3(1 + V —5), (1 + V - 5 ) 2 >

= < 2,1 + >/—5 > < 3,1 + V -5 >

- P P i

and

< 2 > = P 2,

so we choose (guided by the choice in the proof o f Theorem 2.1.1)

A = P l, B = P.

Clearly (A, B ) = 1 and

cB = 1 = < 1 + \ / —5 > < 2 > " * BZ

= P P \P ~2P = P i = A.

I f c e P - is such th a t jiV(c)| > 1 so th a t c ^ 0, by Theorem 2.1.1 there

exist unique nonzero ideals A and B o f Ok such tha t

A = cB, (A, B ) = 1,

and we define by analogy w ith (1.2.2) the arithm etic function a/(c) for any

nonzero ideal I o f Ok by

af (c) = <

0, i f A / I , B / I ,

-N (A ) , if A 11, B / I ,

N (B ), i f A / / , B \ I ,

N ( B ) - N ( A ) , i f A \ I , B 11.

(2.1.4)

2.2 Rademacher’s extension of John’s

theorem

The follow ing extension of John’s theorem was proved by Rademacher [27,

Theorem, p. 173] in 1936.

T heorem 2.2.1. Let M be a nonzero ideal o f Ok - Let c € K be such that

|iV(c)| > 1. I f f ( x ) is Riemann integrable on [0,1] and o f period 1, then

where the sum is over nonzero ideals I o f Ok equivalent to M and the sum

mands are arranged according to increasing N ( I ) .

The above series inherits its convergence from the ordering o f N ( I) . Note

also th a t we only require the function f { x ) to be Riemann integrable on its

period, which is a looser requirement than being of bounded variation.

2.3 Rademacher’s theorem for algebraic

number fields

In th is section, we consider the special case of Theorem 2.2.1 when f ( x ) is

identically 1 and K is assumed to contain an integral element c of norm ±2.

The la tte r condition ensures th a t a j(c) = ±1 for every nonzero ideal I o f

T heo rem 2.3.1. Let K be an algebraic number field such that there exists

c G Ok with |iV(c)| = 2. Let M be a nonzero ideal o f Ok - Then

where the sum is over a ll nonzero ideals I o f Ok equivalent to M and the

summands are arranged according to increasing N ( I ) , and

O k -

(2.3.1)-1 , i f < c > \ I .

Proof. We take

A = < c > , B = < 1 >

so that

c = A /B , (A, B ) = 1.

Also

N (A ) = N (< c > ) = \N(c)\ = 2.

For any nonzero ideal I o f Ok we have

N ( B ) = 1,o j(c) = <

if < c > J ( I ,

N (B ) - N (A ) = 1 - 2 = - 1 , i f < c > | I ,

= —e(c, A).

Also let f ( x ) = 1 (x 6 R). Then f ( x ) is Riemann integrable on [0,1] w ith

Jq f ( x ) dx = 1 and has period 1 as f ( x + 1) = 1 = f ( x ) ( i e R). Hence, by

Rademacher’s theorem, we have

ai{c)E N ( I)= k log 2

so that

E

le [M ]

e(c, I ) 2T+snsR (K )

m m N W w ( K ) J \ m j \

completing the proof.

log 2

□

2.4 Rademacher’s theorem for imaginary

quadratic fields

Let K be an im aginary quadratic field. Then there exists a unique squarefree

integer m < 0 such th a t K = Q (y/m ), see for example [2, Theorem 5.4.1, p.

95]. In th is case

3 II r = 0, 3 = 1 ,*

2, i f m 7 —1, —3,

II 4, i f t o = — 1 ,

6, i f to = —3,


d(K) =m, if to = 1 (mod 4),

0K =

4to, if to ^ 1 (mod 4),

Z + Z a/ to, ifT O = l (mod 4),

Z + , i f to ^ 1 (mod 4),

R ( K ) = 1,

see for example [2, Theorem 5.4.3, p. 98; Theorem 5.4.2, p. 96; D efin ition

13.7.1, p. 380]. The structure constant « of K is given by

7r 4 ’

7T

3 7 3 ’7r

v W ’7T

^ • y /H

if TO = —1,

i f to = -3 ,

i f to = 1 (mod 4), to ^ —3,

, i f to ^ 1 ( m o d 4 ) , T O ^ — 1.

(2.4.1)

Rademacher’s theorem for im aginary quadratic fields, a special case of

Theorem 2.3.1, is as follows:

T heo rem 2.4.1. Let K be an imaginary quadratic field such that there exists

c E Ok with N (c) = 2. Let to be the unique squarefree negative integer such

that K = Q (v/ro). Let M be a nonzero ideal o f Ok - Then

e(c, I )Eie[M] m

= —/clog 2,


mands are arranged according to increasing N ( I ) , e(c, I ) is given by (2.3.1)

and k is given by (2.4.1).


We next determine those im aginary quadratic fields K such th a t O k

contains an element c of norm 2.

I f K = Q (\/m ) w ith m ^ 1 (mod 4) we seek c = x + y \/rn (x, y G Z)

such th a t a;2 + |to|t/2 = 2. I f |m| > 3 there are no integers x and y satisfying

th is equation. I f |m| = 2, th a t is m = —2, we can take x = 0, y — 1. I f

\m\ = 1, th a t is m = —1, we can take x =■ y — Thus we have the two

possibilities

K = Q { y / - i ) , c = 1 + *,

K = Q(y/^2), c= V = 2 .

I f K = Q (y/m ) w ith m = 1 (mod 4) we seek c = ( j j e Z , x =

Cu -4” Imhy2?/ (mod 2)) such th a t -------^—!— = 2, th a t is a:2 + \m\y2 = 8. I f \m\ > 9

there are no integers x and y satisfying th is equation. For \m\ < 8 the

eligible m = 1 (mod 4) are m = — 3 and m = —7. I f m = — 3 the equation

x2 + 3y2 = 8 has no solutions in integers x and y. I f \m\ = —7 the equation

x2 + 7y2 = 8 has the solution x — y = 1. This gives the single possibility

K = Q (v^7 ), c = i i | E Z .

We examine these three possibilities in the next three subsections.

2.4.1 K = Q (v/Z l) , c = 1 + i

W ith K = Q (a /- I) , we have Ok = Z + Z \ / —1 = Z + Z i As h (K ) = 1,

see for example [10, p. 151], Ok is a principal ideal domain (in fact, Ok is a

Euclidean domain, see for example [2, p. 33]). Here by (2.4.1) the structure7T

constant is k = —.4

We choose

c = 1 + i

so that

N(c) = (1 + *)(1 - i ) = 2.

Thus for an a rb itra ry ideal I o f Ok = Z + Z i we have

, 1, i f < l + i> | I ,e (c ,I) =

-1, i f < 1 + i > / / .

We next determine a necessary and sufficient condition for an ideal I o f

Ok to be divisib le by < 1 + i > . As Ok is a principal ideal domain, we have

I = < a + b i> for some a,b 6 Z. Then

< 1 + i > 11 <=>■ < 1 + i > |< a + b i>

l + i \ a + bi

3 c, d € Z such tha t a + bi = (1 + i) (c + di)

•<==> 3 c, d 6 Z such th a t a = c — d,b — c + d

<==> a = b (mod 2).

Thus

e(c, / ) = e (l 4- i, < a + bi > ) = (—l ) a+6.

As h (K ) = 1 every nonzero ideal of Ok is equivalent to < 1 > so we choose

M = < 1 >

so th a t w ith K * = i^ \ { 0 }

[M \ = {cxM C 0 K \<* e K * }

= {(a + bi) < 1 > C 0 K \a, b e Q, (a, b) ^ (0 ,0 )}

= { < a + bi > \a,b e Z , (a, b) ^ (0 ,0 )}.

Hence, by Theorem 2.4.1 we have

V ' (Z 1.)!+b = —— log 2a2 + b2 4 g

/= < a + 6 i>a,6eZ

Now

< a + > = < a' + b'i > a + bi = 6 (a' + b 'i),

for some u n it 0 o f Ok- Since the only units in O k are ±1 , ± i, we deduce

th a t(—i)o+b ^ x (-!)«+ »

a2 + 62 4 a2 + 627^0 a,bGZ

I= < a + b i> (a,fc)^( 0,0)QttbGZ

Hence we have proved the follow ing result.

Theorem 2.4.2.( _ ! ) « *

4 - ; ^ - m 2 8(a, 6)^(0,0)

We emphasize th a t the sum in Theorem 2.4.2 is ordered according to

increasing values of a2 + b2 so tha t

4 4 4 8 4 4 8 ,“ I + 2 + 4 “ 5 + 8 “ 9 + l 0 ------------~ W,0g2-

As a check on our calculations, we derive Theorem 2.4.2 in another way.

Alternate proof o f Theorem 2.4.2. We have

(_ i)°+ » “ „ ( - i ) a+62 ~* a2 + b2 2—i 2-4 a2 + b2

a,beZ n = l a,6GZ(a,b)^(0,0) a2+b2= n

= E in—1 a,6GZ

a2+62= n

as 2 i l2a + b = a + b — n (mod 2).

Now, by a classical result (see for example [17, pp. 115-120], [35]), we have

E i=*£/-4'(

a2

so tha t

X da,bSZ d\na?+b2=n

2 -~> a2 + b2 2— 1 n 2—i I da,beZ n = l d |n

(<z,6) (0f0)

mmd =l e = lVd

, < l 1 \ A 1 1 1- 4 1 - - + - ------- 1 - X + X - T +3 5 J \ 2 3 4

= “ 4 ( 0 (log 2)

= -7T log 2.

as desired. ^

We note th a t Theorem 2.4.2 agrees w ith the fina l form ula in [27] and w ith

the value of 02(1) given in [37, p. 192].

2.4.2 K — Q ( \ / -2 ) , c = V ^ 2

W ith K = Q ( \/—2), we have Ok = Z + Z \ /—2- As /i(A ') = 1 (see for

example [10, p. 151]), O# is a principal ideal domain (in fact, Ok is a


constant is re = — ■=.2 y/ 2

We choose

c = V —2

so tha t

7V(c) = (> /= 2 )(-V = 2 ) = 2.

Thus for an a rb itra ry ideal J o f Ok = Z + Z \ /—2, we have

1 ,if < \/~ 2 > |

1, i f < y p 2 > / I .


O k be to divisible by < y/ — 2 > . As O k is a principal ideal domain we have

I = < a + b^f—2 > for some a, b e Z. Then

< y f— 2 > | / 4=4> < \ /^ 2 > |< a + b \ fm >

y/ — 2 | a + by/ —2

>/—2 | a

4=r> 2 | a.

Thus

e(c, / ) — e (\/—2, < a + by/ ^ 2 > ) = ( -1 ) “ .

As h (K ) = 1, every nonzero ideal of Ok is equivalent to < 1 > so we

choose

M —< 1 >

so tha t

[M \ = {a M C Ok \ol G K * }

= {(a + by/^2) < 1 >C 0 ^ |a ,6 G Q, (a,b) / (0 ,0 )}

= {< a + &>/—2 > |a, 6 G Z, (a, 6) ^ (0 ,0 )}.

Hence, by Theorem 2.4.1, we have

/#< o>/=<a+6's/~2>

a,6€Z

(~1)° = l _ l 0g2^ a2 + 2 b2 2 ^2

Now

< a + by/— 2 > = < a7 + b'yf— 2 > <£=>■ a + by/ — 2 = 6 {a! + 6z\Z~2),

for some u n it 0 o f Ok = Z + Z \ /—2- But the only units o f Z + Z%/—2 are

±1 so

V ( - 1)* 1 V ( - 1 ) °

. A ; a2 + 2 ^ 2 4 - i a2 |2 6 2'/#< 0> a,6ez/=<a+fcV—2> (a,6) (0,0)

a,b€Z


T heo rem 2.4.3.

(a ,6 ) /(0 ,0 )


As a check on our calculations, we derive Theorem 2.4.3 in another way.

Alternate proof o f Theorem 2.4.3. We have (as a = a2 = a2+2&2 (mod 2))

(-1)° = ~ v (~l)a4"1 a2 + 262 " 4 - a2 + 262

a,i>6 Z n = l a,6GZ(a,6)^(0,0) a2+262= n

= E ^ r E i-n = l o,6GZ

a2+262=7i

Now

E ! = 2E (t )a,6eZ d|n ' '

a2+2b2=n

(see for example [11, Theorem 64, p. 78], [36]) so th a t

( - l ) r t * = ~ ( - i ) . / _ 8 -

+ 262 Z -. n d .(

(_ l)de / _8

a,6G Z 7 i= l d|n(a, 6)^(0,0)

^ ^ de V da,e= l N

= 2 E( - l ) de / —8'

de \ d Ja ,e= l x '

-e = l d = l

= -2 (lo g 2 ) L ( l, -8 ) ,

where the D irich le t L-series L ( l, D ) for an a rb itra ry discrim inant D is given

by^ ( - ) t(l,D) = E iJ2-n ~ \


For D < 0 by D irich le t’s class number form ula we have

■ A (g ) 2tth(D )^ n w (D )-J \D \'

see for example [17, Theorem 10.1, p. 321]. W ith D = -8 , as w (—8) = 2,

\J \D \ = y/ 8 = 2y/2 and h (—8) = 1, we have

f ' S ) = - J Ln 2^2"71=1 V

Thus

T (~ 1}‘ - 2floC2) * - ’rl0g2 ** + » 2v*2 V2 '

(a,6 )^ (0 ,0)

We note th a t Theorem 2.4.3 agrees w ith the value of 0 i( \/2 ) given in [37,

p. 192].

W ith K = Q C v^T), we have 0 K = Z + Z ( i± ^ ) . As h (K ) = 1 (see

for example [10, p. 151]), Ok is a principal ideal domain (in fact, Ok is a


constant is k = —=.V7

We choose1 + a/= 7

C = ^ —

so tha t

Thus for an a rb itra ry ideal I o f Ok = Z + Z ^ 14Y ~ ^ , we have

l , i f ( i ± # z ) | / ,e(c, I ) =

_ l , i f


Ok to be divisib le by *s a Pr inciPal ideal domain we

have I = (^a + b |

<i±r EZ>

1 +' j for some a, b G Z. Then

( !± * 2 >1 + y/—7

21 + y/ ~ 7

a + b

• * * m >

t 3 )

1^3) (1^3)

2 I a.

a

a

Thus

e (c ,/) = e


choose

M = < 1 >

so tha t

As

[M \ = {a M C Ok W € K * }

= {(z + y V ^ f ) < 1 >C 0*1®, y e Q, (x, y) ^ (0 ,0 )}

= | / a + 6 f l ^ - I ) \ k 6 e Z , ( a , 6 ) ^ ( 0 , 0 ) j .

= ( a + & ( i J ^ E Z ) ) (a+&(lz EZ))— o? 4* ob -4- 2 ,

by Theorem 2.4.1 we have

/ - iv *= — 7= log 2.y - ( ~ l ) a ____

a2 + ab + 2 b2 y/ 7i^< o>

a,&€Z

Now

„ . ( l ± £ 1 ) .

where 0 is a u n it o f Ok = Z + Z . B ut the only units o f Z + Z

are ±1 so

v ( ~ l) a _ 1 v (~ 1)arT~Z a2 + ab + 2 b2 2 ^ a2 + ab + 2 b2'/#< o> a,bez

J = ( a + b ( ^ V Z?) ) (o.6)#(0,0)a,e>€Z


T heorem 2.4.4.

V ' (~ 1}° - log 22 , a’ + a& + 26> V7

(a,6) (0,0)

As the pa rity o f a is not d irectly related to the parity of a2 + a6 + 262, i t is

not possible to give a proof o f Theorem 2.4.4 along the lines of the alternative

proofs of Theorems 2.4.2 and 2.4.3 given earlier.

2.5 Rademacher’s theorem for real quadratic

fields

Let K be an real quadratic field. Then there exists a unique squarefree

integer m > 0 such th a t K — see for example [2, Theorem 5.4.1, p.

95]. In th is case

n = 2, r = 2, s = 0,

w (K ) = 2,

{ m, i f m = 1 (mod 4), v

4m, i f m ^ 1 (mod 4),

{Z + Z yfrri, i f m = 1 (mod 4),

Z + z ( ± ± ^ ) , i f m ^ l (mod 4),

R (K ) = log rj, where 77 is the fundamental u n it(> 1) o f AT,


see for example [2, Theorem 5.4.2, p. 96; Theorem 13.7.1, p. 380]. Thus, as

Rademacher’s theorem fo r real quadratic fields, a special case of Theorem

2.3.1 is as follows:

T heo rem 2.5.1. Let K be a real quadratic field such that there exists c & Ok

with |iV(c)| = 2. Let m be the unique squarefree positive integer such that

K = Q (y/m ). Let M be a nonzero ideal o f Ok - Then


mands are arranged according to increasing N ( I ) , e(c, I ) is given by (2.3.1)

and n is given by (2.5.1).

D e fin itio n 2.5.1. An integer a o f areal quadratic field is said to be prim ary

the field.

T h eorem 2.5.2. Every nonzero real quadratic integer has exactly one asso

ciate which is primary.

i f m ^ 1 (mod 4).

(mod 4)(2.5.1)

a > 0, I < —7 < r f , a

where a ' denotes the conjugate o f a and rj (> 1) is the fundamental un it o f

Theorem 2.5.2 is proved in Cohn [10, Theorem 3, p. 146].

In Section 2.4 we saw th a t there are only three im aginary quadratic fields

K containing an integral element c o f norm ±2. In contrast each of the

in fin ite ly many real quadratic fields Q(y/p) (p (prime) = 3 (mod 4)) contains

such an element.

2 .5 .1 K = Q ( \ /2 ) , c = a/2

W ith K = Q (\/2 ), we have Ok = Z + Z \/2 - As h (K ) = 1 (see for example [10,

p. 271]), Ok is a principal ideal domain (in fact, Ok is a Euclidean domain [2,

Theorem 2.2.6, p. 35]). The fundamental u n it is ift = l+ \ /2 (see for example

[2, Example 13.7.1, p. 381]), so the regulator is R (K ) = lo g (l + \/2 ). Here

by (2.5.1) the structure constant is k =

We choose

c V2

so tha t

N(c) = (V2)(-\/2) = -2.

Thus for an a rb itra ry ideal I o f Ok = Z + Z \/2 , we have

eM)J i . * < V 2 > | / ,

\ - l , i f (V2)XI .


Ok to be divisib le by (\/2 )- As Ok is a principal ideal domain we have

I = < a + by/2 > for some a, 6 6 Z. Then

< V 2 > \ I <=$> < V 2 > \< a + by/ 2 >

y/ 2 | a 4- by/ 2

V 2 | a

2 I a.

Thus

e(c, / ) = e(V2, < a + 6\/2 > ) = (—l ) a.


choose

M = < 1 >

so tha t

[M ] = {a M C O k\& € AT*}

= {(a + &V2) < 1 > C 0 K\a, b € Q, (a, 6) / (0 ,0 )}

= | < a + by/2 > |a, 6 € Z, (a, b) / (0,0). jAs N (< a > ) = |Af(aOI> we have

N ( < a + by/2 > ) = |JV(a + &V2)|

= | (a + 6 \/2 ) (a — by/2) |

= |a2 — 262|,

and so by Theorem 2.5.1, we have

V ,(-!)“ = _ M 1.±.,V2)1 2l“ 2 - 2^1 V2 l0g2-

/=<a+b\/2>a,beZ

We next determine a unique generator a+by/2 for the ideal I . By Theorem

Let p (prime) = 3 (mod 4) and le t K = Then Ok = Z + Z^/p.

We note th a t Ok is not necessarily a principal ideal domain. Here we have

n = 2, r = 2, s = 0, w (K ) = 2, d (K ) = 4p and R (K ) = log rj, where rj is the

fundamental u n it of K .

I f p — 3 (mod 8) then x2 — py2 = - 2 is solvable in integers x, y, and if

p = 7 (mod 8) then x2 —py2 = +2 is solvable in integers x, y, see for example

[2, Exercises 14,15, p. 297] respectively. This gives rise to an element o f norm

(—l ) E7i 2 in K = Q (y/p) whenever p = 3 (mod 4). Let c = x + y^Jp £ O k

2.5.2 we set a + by/2 in the range

a + W 2 > 0 , 1 < a + b ^ < ( 1 + a / 2 ) 2 ,a — by/ 2

th a t is,

a + bV2 > 0, 1 < < 3 + 2 ^ ,a — by/ 2

to get a unique generator.


T heo rem 2.5.3.

a,be Za+b'v/2>0

2.5.2 K = Q (y/p), p (prime) = 3 (mod 4)

be such an element of norm (—1 )^ 2 . As x2 — py2 = (—1)£412 we have

x2 + y2 = 2 (mod 4) so th a t x = y = 1 (mod 2).

Let I be a nonzero principal ideal of Ok - Then, by Theorem 2.5.2, there

exists a unique element a + by/p o f Ok such th a t I = < a + by/p > and

a + by/pa + by/p > 0 , 1 <

We have

e(c, I ) = e(c, < a + by/p > ) = <

Now, as x = y = 1 (mod 2), we obtain

a - b y / p < V

1, i f c | a + by/p,

- 1 , i f c / a + by/p.

so tha t

Also

c | a + by/p 4=» x + yyfp | a + by/p

2 | (a + by/p)(x - yy'p)

2 | (az - pby) - (ay -

4= ^ 2 \ (a + b) - (a + b)y/p

2 I a + b

e(c, / ) = e(x + y y /p , < a + by/p > ) = ( - l ) a+6.

N ( I ) = N ( < a + by/p > ) = |iV(o + by/p) \

= \ (a + b y / p ) ( a - b y / p ) \ = \a2 - p b 2 \.

T heorem 2.5.4.

(_l)a+f> (log 2) (log rj)

S I VPa+by/p> 0

2.6 Rademacher’s theorem for real cubic fields

with two nonreal embeddings

Let K be a real cubic field w ith two nonreal conjugate fields. Then n = 3,

r = 1 and s = 1. The only roots of un ity in K are ±1 so w (K ) — 2 (see

for example [2, Theorem 13.5.3, p. 367]). The cubic field K has a unique

fundamental u n it 77 (> 1) (see for example [2, Theorem 13.4.2, p. 362]). The

regulator R {K ) of K is

R (K ) = | log 7 7] = log 77,

(see for example [2, D efin ition 13.7.1, p. 380]). The structure constant o f K

2r+BirsR (K ) 2tt log r/

w (k ) V \ W T \ V W M '

T heorem 2.6.1. Let K be a real cubic field with two nonreal embeddings.

Letr) > 1 be the unique fundamental un it o f Ok , so that a ll units o f Ok are

given by ± 77™ (71 € Z ). Given a £ Ok \ { 0} there exists a unique associate f i

o f a such that

13 > ° ’ 1 ~ \NQ3)\V2 < V'

Proof. The associates o f a are ± rjna (n € Z ). I f a > 0, we choose /3 =

+r]na > 0, and i f a < 0, we choose (3 = —r)na > 0. Thus /3 is a positive

associate of a.

Clearly

P = \P \ = Vn\<*\ = sgn(a)77n.

Denoting the conjugates o f (3 by 0 and 0 ' , we obtain

0 = ± (V ) a , 0 ' = ± a .

Hencea' a" ( ' " \ n ' "f3 (3 = ( e e ) a a .

As 77 is a u n it o f Ok we have

777/ 77” = A (77) = ±1.

Now 77 > 1 and 77" = r f so th a t 0 0 ' = 0 0 = \0 12 > 0 and thus

' " 17777 77 = 1.

Hence

Thus

and so

/ / / 1 ” = v

77n

pprpr

P \a\rj2n

\P'P"\ K aw|

|o;|7y2rlNow ; „, is a s tric tly increasing function of n G Z as u > 1 such tha t

|o; a I

.. \a\rpn \a\rj2nhm \ - r - a 7 = 0. 1™ Tr~ n7 = + 00.n~ *~ 00 |q: a I n-»+oo | a a |

Hence there exists a unique m G Z such tha t|o ; |^ 2 ( m - l ) ^ ^ |o ; |^ 2m

th a t is

/ 111 ^ l — i / n 1 >a a cc a

| a aThus 0 = sgn(a)r}n is the unique associate of a such tha t

00>O, 1 <F ina lly we note tha t

0 0

01 0

< 1 f .

so th a t the condition 1 <

0 0 " 0 0 0 " N (0 )

- K < rj2 is equivalent to

1 <~ \N (0 )\W V

giving the asserted result. □

T h e o rem 2.6.2. Let K be a real cubic field with two nonreal conjugate fields.

Suppose that there exists c G O k with |iV(c)| — 2. Then

e(c, I ) 27r(log7?)(log2)

V N ( ! ) s / W ) '

where the sum is over a ll nonzero principal ideals I o f Ok and the summands

are arranged according to increasing N ( I ) , and e(c, I ) is given by (2.3.1).

We now apply Theorem 2.6.2 to the cubic fie ld K = Q (v/,2) which is

a real cubic fie ld w ith two nonreal conjugate fields Q (w s/2) and Q(tu2s /2),

where w G C is such th a t w ^ 1, w3 = 1.

2.6.1 K = Q ( \//2), c = \/2

Let K = Q (s/2). Then d (K ) = —108, see for example [2, Table 1, p. 177].

As h (K ) = 1 (see for example [2, Table 9, p. 329]), O k = {u + v \ / 2 +

vj{y/2)2 \u ,v ,w € Z } is a principal ideal domain. Let c = \/2 G Ok- Then

N(c) = 2. We have n = 2, r = 1, a = 1, w (K ) = 2, d {K ) = -108. The

fundamental u n it is rj = l + \ / 2 + ( \ /2 ) 2 [2, Table 11, p. 375], so the regulator

is R {K ) = lo g (l + $ 2 + (V 2 )2). Hence

27rIog(l + ^ + ( ^ ) 2) 7rlog(l + V 2 + ( V 2 ) 2)K V I -1 0 8 ] 3 \/3

Let I be a principal ideal of Ok - Then, by Theorem 2.6.1, there exists a

unique (5 = x + y \ / 2 + z(ty2)2 G Ok such th a t I = < (3 > and

/ 3 > 0 , 1 - |iV(/?)|V2 < r >

th a t is,

x + y + z{\J2)2 > 0 , 1 <x + y V2 + z { $ 2 ) 2

< 1 + V2 + (v ^ )(x3 + 2 y3 + 4 z3 — 6 xyz ) 1! 2

as

N ^ + y y f i + z \ f 2 = a:3 + 2y3 + 4z3 — 6 xyz > 0.

We have

e(c,J) = e (c ,< /3 > )

1, i f < c > |< j3 >

-1 , i f < c > /< 0 >


= ( - l ) x-


T heo rem 2.6.3.

( -1 ) *

1, i f c | 0 ,

-1 , i f c //3 ,

1, i f v^2 | x + y f f i + z ( ^ 2 )2,

-1 , i f y/2 )(x + y ^ / 2 + z ( ^ 2 )2,

1, i f v^2 | x,)

-1 , i f \^2 )(x,1

1, i f 2 | x,

-1 , i f 2 /x ,

E „ x 3 + 2 y3 + 4 z3 — 6 xyzx,y,z€Z y y

x+yfy2+z{fy2)2>0

7r(log 2) lo g (l + y/2 + ( y ^ )2)

3 \/3


Chapter 3

Kronecker’s Theorem

3.1 Dedekind eta function

In th is section we give some properties of the Dedekind eta function th a t we

w ill need in the next section.

Let H denote the upper ha lf of the complex plane, th a t is

H — {z G C\z = x + iy , x ,y eM , y > 0}.

For z e H the Dedekind eta function 77(z) is defined by

OO

ij(z ) = eKiz/12 [ I ( ! “ e2” imz). (3.1.1)771=1

This in fin ite product converges absolutely and uniform ly in every compact

subset of H , see for example [32, p. 15]. Thus i](z) is analytic in H . Since

86


CHAPTER 3. KRONECKER’S THEOREM 87

none of the factors o f the convergent in fin ite product is zero for z € H it

follows th a t r)(z) ^ 0 for z € H , see [32].

Theorem 3.1.1.

\rj{x + iy)\ = \r } ( -x + iy)\

fo r all x, y € M with y > 0.

Proof. Let x, y G K. w ith y > 0. Then

OO

r)(±x + iy ) = J J ( l - e2nim ±x+i^ )m= 1

oo= n a - e~2nmye±2nimx)

m= 1

so tha t

\r j(±x + iy)\ = e~ ] ^ [ | l -oo

I - e -*nu>ye,

m= 1

Now| j g—2nyg2m m x | _

so tha t

^ g—2iry g2irimx _ I j g —2irj/g—27rimi I

\r)(x + iy)\ = \r } ( -x + iy)\,

as asserted. □

The fundamental transform ation formulae of r/(z) are [32, pp. 17-18], [34,

Vol. 3, p. 113]

7)(z + 1) = eni/12rj(z), r) = \ /^ iz r ) (z ) , (3.1.2)


where the branch o f the squareroot is taken so th a t i t has the value 1 at

z — i.

I t follows from (3.1.2) th a t i f a ,p ,y , 8 £ Z satisfy aS — P7 = 1 then

77 O /g + f ) = ev V + < ^ (*), (3.1.3)

where e = e (a ,P ,y ,8 ) and |e| = 1, see [32, Prop. 3, p. 17].

Hence

V'a z + jT ,7 z + 8 , = h 'z + ^ M z ) ! 4 (3.1.4)

Now le t ax2 + bxy + cy2 and a'x2 + b'xy + dy2 be two positive-definite,

integral, binary quadratic forms of discrim inant d (so th a t a, a' > 0, c, d >

0, d < 0) which are equivalent, so th a t there exist a ,P ,y ,5 G Z w ith aS —

Py = 1, say

a'x2 + b’xy + dy2 = a(Sx + Py) 2 + b(8 x + Py)(yx + ay) + c( 7® + a y)2.

Then

and

Set

Then

a' = aS2 + bSy + cy2

b' = 2a8P + byP + b8 a + 2cya.

b + V d

az + p _ b ' + yfd yz + 8 2 a'

G H


and

Thus, by (3.1.4), we have

\y z -\-8 f = —.a

4a' fb + V d \

n * J a H * j (3.1.5)

We note tha t

rj(iy) € E+, e ~ ^ - q e E + (3.1.6)

for y > 0, so th a t

V'1 + iy '

(3.1.7)

E xam p le 3.1.1. We consider the form x2 + 2y2 o f discrim inant —8. Here

a = 1,6 = 0 ,c = 2,d = —8. As x2 + 2y2 is equivalent to 2x2 + y2 (since

2x2 + y2 = (Ox + (—1 )y ) 2 + 2 ( lx + 0y2)) we have a' = 2, b' = 0, d = 1, d' =

d = — 8 . 77ms by (3.1.5) we hove

48 \

4

H 4 j = 2" { 2 )

Then by (3.1.7) we deduce

Taking logarithms o f (3.1.5) we obtain

'b + y/dlog a — 4 log V

2 a= log a' — 4 log V

V + \ fd2a!

(3.1.8)



In particular, as ax2 + bxy + cy2 and cx2 — bxy + ay2 are equivalent forms

(since ax2 + bxy + cy2 = c(0x + 1 y ) 2 — b( Ox + ly ) { —lx + 0y) + a(—lx + 0 y)2),

we have by (3.1.8) and Theorem 3.1.1

log a — 4 log Vb + y/d'

2 a= log c — 4 log

= log c — 4 log

V

V

'- b + Vd2c

’b + Vd2c

. (3.1.9)

We close th is section by noting the follow ing two im portant relationships

satisfied by the Dedekind eta function, which follow from the theory of theta

functions, see for example [18, p. 275]: for ^ € H

' 1 + z 'V (!) rj(2z) = em/24if(z),

+ lG r)(2zf = e ~ ^ 3rj’1 + z'

(3.1.10)

(3.1.11)

3.2 W eber’s functions

For z e H Weber’s functions f (z ) ,f i(z ) ,f2(z) are defined in terms o f the

Dedekind eta function by

f(z) = v ,

*?(§)h i? ) =rj(z) ’

f2(z) = V2rj{z) '■

(3.2.1)

(3.2.2)

(3.2.3)



see [34, Vol. 3, p. 114]. From (3.1.10), (3.1.11), (3.2.1), (3.2.2) and (3.2.3)

we obtain

(3.2.4)

and

/ ( * ) ' = A M 8 + / 2(* )8 (3.2.5)

see for example [34, Vol. 3, p. 114].

From (3.1.6) and (3.2.1) - (3.2.3) we see tha t i f m 6 N then f ( y /—m),

/itV^m), e R+.

E xam p le 3 .2.1. Prom Example 3.1.1 and (3.2.2) we deduce

in agreement with the value given in [34, Vol. 3, p. 721].

3.3 Kronecker’s limit formula

Throughout th is section, ax2 + bxy+cy2 is a positive-definite, integral binary

quadratic form of discrim inant d = b2 — 4ac.

We saw in Theorems 2.4.2, 2.4.3 and 2.4.4 th a t Rademacher’s theorem

allowed us to evaluate the sums

M V = 2 ) = 21' 4

(m ,n )^( 0,0)m ,n€ Z

(m ,n )^( 0,0)m,n€Z

(m,n)^(0,0)

m 2 + m n + 2n2'( - 1)


However i t does not seem possible to use Rademacher’s theorem to evaluate

the more general sums

“ , ( - 1) " y V ( - 1)" W ( ~ l) " +n^ am2 + bmn + cn2 ’ am2 + bmn + m 2 ’ am2 + bmn + cn21

m ,n = —oo m ,n = —oo m ,n = —oo

where the dash (') indicates th a t the term (m, n) = (0,0) is om itted. For

these we use a theorem o f Kronecker, known as Kronecker’s lim it formula,

which asserts th a t as s —* 1+ we have0°

V ' r— 5— ^ ----------j r - = • — !— + K (a ,b ,c) + 0 ( s - l ) , (3.3.1)^ (am2 + bmn + cn2)s . / } s — 1

771,71=—OO V ' V I I

see for example [32, Theorem 1, p. 14], where d — b2 — 4ac < 0 and

T.. , . 47T7 2 tclo g \d\ 27r . 87r ,

J f(o ’ 6' c) = ^ f - W + ^ log a ^ log

b + y/d^ “ 2 7

. (3.3.2)

By (3.1.9) we see th a t

K (a ,b ,c ) = K (c,b ,a ). (3.3.3)

00Theorem 3.3.1. (i) £ - i + = 2g(4a, 2i», c )-g (a , 6, c),

771,71=—OO(m ,9l)^(0,0)

771,71=—OO(171,71)^(0,0)

^ ( i ) m+n 27Tlog4 ,(m) > — ; r = ------- 7= ----- 2 A (4a, 26, c)

^ am2 + bmn 4- cn2 */OF771,71=—OO V I I

(m ,n )^(0 ,0 )

—2K(a, 26,4c) + 2Ar(a, 6, c).


Proof, (i) Let s > 1. Then

( i + ( - i nY ' u + i - i n = y '' (am2 + bmn + cn2)s ^ (am2 + bmn + cn2)s

n = —oo N 7 m ,n = —oo v 7m even

oo ^

= E' (4am2 + 2 bmn + cn2) 3 *m ,n = —oo v 7

Hence, as 4am2 + 2bmn + cn2 has discrim inant

(26)2 — 4(4a)c = 462 — 16ac = 4(62 — 4ac) = 4 d,

we obtain, by Kronecker’s lim it formula

(~ l)m______^ (am2 + bmn + cn2) 3m ,n =—oo 7

oo oo

= 2 V ' _________ -_____________ V '_ -________(4am2 + 2 bmn + cn2) 3 (am2 + bmn + cn2 ) 3

m ,n = —oo N 7 m ,n = —oo v 7

= 2 K (A a ,2 b ,c )-K (a ,b ,c ) + 0 ( s - l ) .

Letting s —> 1+ we obtain

00 > ( 1lim V t , >■ ;--------- r r - = 2K(4at 2bt c ) - K ( a , b t c).

s_>1+ (am2 + bmn + cn2) 3 v ’ ' y 5 ’ ’m ,n = —oo 7

°° > (—i ) mAs > t— -— --------------— converges, we obtain the asserted result.

' (am2 + bmn + cn2)m ,n = —oo v 7



(ii) We have by part (i) and (3.3.3)

y ( - i ) n = y ______( - i ) mam2 4 - hm.n. A- m? Q/p -]

o

- E ( - 1 ) ’

am? + bmn + cn2 ^ an2 + bnm + cm?m ,n=—co m ,n = —oo

cm2 + bmn + an2m ,n = —oo

= 2K (4c, 26, a) - K (c, 6, a)

= 2i f (a, 26,4c) — i f (a, 6, c).

(iii) Let s > 1. We have

y (1 + ( - ! ) " ) ( !+ (-!)") = y V 4(am? 4- bmn + cn2)s ^ (am? + bmn + cn2) 6

m ,n =—oo N ' m ,n = —oo ' 'mtn even

1 00 — Y 'As~l *—J (am? + bmn 4- cn2)sm ,n = —oo K 7

i

so tha t

y = f_}_ _ 4 y - ' 14— (am2 + 6mn + cn2)5 I 4s-1 / (am2 + bmn + cn2) 3m ,n = —oo ' / \ / rn,n=—oo ' 7

( - 1)«00- y ________

(am2 + bmn + cn2)s77l,n=—OO v 7uu

- E' ( - i ) "' (am2 + 6m n 4- cn2)6

,71=—OO x 7

Now

— i = g-C3- 1)1 4 _ i43~i

= (1 — (s — 1) log4 + 0 ((s — l ) 2)) — 1

= - ( s - 1) log 4 + 0 ((s - l ) 2)



so tha t

1 \ 00

£4 3 - I * J ^ ( a m 2 _|_ f} m n 4 . o t 2 ) 3771,71=—00 v '

(-(> - 1) log4 + 0 ((s - l)2)) ( , 1 , + /s'(a, 6, c) + 0 (s - 1)

—27rlog4>/MI

+ 0(s — 1).

Hence le tting s -+ 1+ and appealing to (i) and (ii) we obtain

(_ l)m +n — 27T log 4£ '

7711 Tl - OOam2 + 6mn + cn2 - ^ ~ W-*,2f,,c) - c))

—(2 if (a, 26,4c) — K (a, b, c))

as asserted. □

Appealing to Theorem 3.3.1 and (3.3.2), we obtain the follow ing theorem.

771,71=—OO

Theorem 3.3.2. (i) ^

m f r (-1)771,71=—C

OO

(«o £ '

( - 1)" 87ram2 + bmn + cn2

log / l'b + Vd'

2 a

87T

771,71=—OO

am2 + bmn + cn2

( _ ^ ’w+n

am2 + bmn + cn2

v f f l

87T

log

log

/2

/

'& + V 3 '2a

6 + Vd.2a

Proof, (i) By (3.3.2) we have

K (a ,b ,c ) = - B lh iM + - - ^ l o gs /R vW v13f v/RI

Vb + -\/d

2 a



and

K(4a, 2b, c) =47T7

~ 7W \27T7

27r log 4 |c?| 27rlog4a 87T

v / iR V * Rlog V

'2 b + y/4d 8 a

7rlog4 7rlog |d| 7rlog4 7rloga

V R [ y jd f v13f \ /P i y p f47T

log Vb + Vd

4 a

27T7 7T log |d| 7T log a 47rlog

b + Vd4 a

so tha t

2 i^ (4a, 2b, c) — i f (a, b, c)

47T7 27r log |d| 27r log a 87r

a /R V P i -Mlog V

b + Vd4 a

47T7 2n log \d\ 27rloga 87r ,H 7 = ----------------- 7 = = - 7 = log

87T

W i

■ M VW\^b±& '\

VW \ V W \V

b + Vd2 a

logV \ 2cT)

( b+Vd\{ 4a )

Then, appealing to Theorem 3.3.1 (i) and (3.2.2), we obtain

£ ( - i ) ’772,73,=—OO

(m ,n)^(0,0)

am2 + bmn + cn28?r

V W \

87T

~ W \

logV (*8?)V

log

( b+\/d\V 4a ;

b + Vdf i 2 a

(ii) By (3.3.2) we have

47T7K (a , b, c) —

27rlog|d| 27rloga 87r , —7= = ---------------- + ------; = --------- 7= log 77v ld f v ld i v W V H

'6 + V d '2a


and

K (a , 2b, 4c)47T7 27T log 4|d| ^ 27rloga 8 tt

vipf , / mlog

27T7 7rlog4 7rlog\d\ ^ 7rloga 47r

v f f l v W vTdf v M >/PT

V

log

2 6 + \/4 d ' 2 a

b+ VdV '

so tha t

2 K (a , 2b, 4c) — K (a , b, c)

47T7 27rlog4 27rlog|d| 27rloga 87r+

% / H v W % / H v ' M l v f f llog V

'b + Vd

47T7 ^ 27T log |d| 27r log a ^ 871" ^

v W y p f v r i y R

47r log 2 87r

V'b + Vd'

2 a

log■n { < ¥ )

V (I 2a )

Then, appealing to Theorem 3.3.1(ii) and to (3.2.3), we obtain

( - 1)

m .n —~ooam? + bmn + cn2

4 7 r , _ o 8 ? r I

\ m v w \

87r , „ ( b + Vd

V' b + \/d \ < a )

Vf in -v tAI 2a )

log f i2 as/¥\

(iii) Prom parts (i) and (ii) we have

-2tf(4 a , 26, c) - 2 t f ( a ,2b, 4c) + 2K (a , b, c)

= —(2i^(4a, 26, c) — K (a , b, c)) — (2K (a , 2b, 4c) — K (a , b, c))

87r

V W \log

Vb ± V d \

4 a )

47t log 2 8 n

VW \ V\d\log

( te a )

= 27T log 4 8tr I7? ( ^ ) 117? C +4 ^ )

V P f "s/Pi

Appealing to Theorem 3.3.1(iii), and to (3.2.2), (3.2.3), and (3.2.4), we

obtain

£771,71=—OO

(m ,n)^(0,0)

am2 4- bmn + cn287T

7 R

87T

log 2 2

log

Ab + Vd

2 a

V 2

AW V d '

2a

- l

87T

V\d\log / b + V ]d [

2a

completing the proof. □

In the next theorem we reprove Theorem 2.4.3 using Theorem 3.3.2.

T heo rem 3.3.3.

f * ( - l ) m2—* m 2 + 2n2

771,71=—OO(m ,n)#(0,0)

V 2 l0g2 '

Proof. From Example 3.2.1 we have

h (v^2) = 21/<4.

Taking a = 1, b = 0, c = 2, d = b2 — 4ac = —8 in Theorem 3.3.2 (i) we obtain

t771,71=— OO V

- - t H 21/4>

~ 7 I los2as required. □

Taking a = 1, b = 0, c = A e N, so tha t d = b2 — 4ac = —4A < 0, in

Theorem 3.3.2 we recover a result of Zhang and W illiam s [37, Theorem 3, p.

189] proved in 1999. We give the results in terms of Ramanujan’s functions

3k = 2 -1' 4/i(v /=A), Gj = 2 -1'7 (v '= A ),

see [29, p. 27].oo / ' i \m

T h e o rem 3.3.4. (i) £ log (2 *}) .771,71=—OO v

(m ,n)^(0,0)

( ~ l ) n 7T

771,71=—OO(m ,n)^(0,0)

(“) E rf+w = _ log(2G*)771,71=—OO v

(m ,n)^(0,0)

Proof, (i) By Theorem 3.3.2(i) we have

(-1)™ _ 8tt

rm ,n = —ooE' O T -

4?r log/i(V^A)VA

47f log(21/45A)Va

* log(2).V a


(ii) By Theorem 3.3.2(ii) and (3.2.4) we have

'sAJ

£ 'm ,n = —oo

( - i ) " m 2 + An2

47T7x47T

logV a & / ( vC A )/i (x/= A )

log (2 -V V ( v^ A )2 -1',V i (vc A ))

^ lo g f e G i )

= ^ l o g ^ G j )

(iii) By Theorem 3.3.2(ii) we have

£ 'm ,n =—oo m2 + An2

8?r

2\/A 47T^ lo g / ( v ^ X )

^ l o g ^ C * )

^ lo g ( 2 G y ,

completing the proof. □

The firs t four values in Table V I of [34, Vol. 3, p. 721] are

= 21'4,/ , (V = 2 ) = 2*/*


CHAPTER 3. KRO NECKER’S THEOREM

/(> /= 3 ) = 21/3,

f i (V —4) = 23//8 (w ith a typo corrected).

From (3.2.4) and (3.2.5) we deduce

A ( 7 = 1 ) = h ( 7 = 1 ) = 2 V 8

/ (7=2) = 21'8 (72 + l ) 1/8, A (7=2) = 21'8 (75 - l) V8,

A (7=3) = 21'18 (2 + 7s)1/8 A (7=3) = 2^12 (2 - 73 ) 1/8

/ (7=4) = 2‘/« (1 + 72)1/4 A (7=4) = 2^le ( l - 72) V4

HenceSi = 2-V8, G l = 1,

\ 1/892 1, G2 = 2- 1/8 ( ^ + l ) ,

93 = 2-1/6 (2 + ^ ) 1/8, G3 = 21//12,

f t = 2^ , G4 = 2-3/ 16 ( l + V ^ )1/4.

Appealing to Theorem 3.3.4 we obtain the follow ing series evaluations.

T heo rem 3.3.5.

( - l ) m _ 7TOO

E - ~ > * 2’m 2 + n 2 v.771,7l<— OO

(m ,n)#(0,0)

^ ( — l ) n 7T ,^ m2 + n 2 ~ ~ 2m,n=~oo

(m ,n)^(0,0)

00 / 1 \m + n

2 1 2 = — 7r l ° g 2 -td ." -1- 77"m 2 + n2771,71=—OO

(m ,n)#(0,0)

CHAPTER 3. KRONECKER’S THEOREM

Theorem 3.3.6.

” J _ 1 )m _ _ L l0o;2^ m? + 2 n 2 V2

771,n = —oo v(m ,n)^(0,0)

( “ I ) ” _ ^ l n ^^ m 2 + 2n2 “ ~ 2 \ / l 2 \ / Im ,n =—oo v v

(m ,n)^(0,0)

“ (_!)■»*■ _ ____ j _ „

J iz L * , ™2 + 2" 2 2^ 2 2V2(m ,n)^(0,0)

Theorem 3.3.7.

(—l ) m _ 7T 0 7r

771,71=—OO v v(m ,n)#(0,0)

Y '' ( ~ l) n ^ ^ i

m2 + 3»2 _ " S v ? g + 2VS771,71=—OO v v

(m ,n )/(0 ,0 )

~ ( _ ! ) m + n 4?r

^ m 2 + 3n2 “ 3v/3771,71=—OO V

(ro,n)^(0,0)

Theorem 3.3.8.

_ ( - ! )"» _ 3 f i j^ m 2 + An2 4 g ’

771,71=—OO(m ,n)^(0,0)

E ( — l ) n 7T 7T

m 2 + 4n2 ~ ~ 8 ® 2 ®771,71=—OO

(m ,n)^(0,0)

« ( _ ! ) « + » 7T n 7T ,

m 2 + 4n2 “ 8 g 2 S771,71=—OO

(m ,n)^(0,0)

log (\/2 + 1)

log (\/2 + 1)

5g (2 + V 3 ) ,

5g (2 + \/§ ) ,

I + V 2) ,

i + V2).


3.4 Final results

The evaluation o f Weber’s functions \f(z )\, \f i(z )\ and 1/ 2(2)! at quadratic

irra tiona lities z — (d < 0) has recently been carried out by Muzaffar

and W illiam s [26]. P u tting the ir evaluation together w ith Theorem 3.3.2, we

obtain the evaluation o f the in fin ite series

y V (■-1)™ y V ( -1 ) " y V (-1)™ +"^ am2 + bmn + cn2 ’ am2 + bmn + cn2 ’ ^ am2 + bmn + cn2 ’

m ,n = “ 0 0 771,71=—oo 771,71= —oo

for a positive-definite, prim itive, integral binary quadratic form ax2+ 6 xy+ q /2

of discrim inant d (< 0).

In order to state the result of Muzaffar and W illiam s it is necessary to

introduce some notation. Let d be the discrim inant of a positive-definite,

prim itive , integral b inary quadratic form ax2+bxy+cy2 so th a t d = b2 —4ac =

0 or 1 (mod 4), a > 0, c > 0, d < 0. The conductor / o f d is the largest

integer such th a t d / f 2 = 0 or 1 (mod 4). We set A = d / f 2. The set

o f classes of positive-definite, p rim itive , integral, binary quadratic forms of

discrim inant d(< 0) under the action of the modular group

r ,s , t ,u € Z, ru — st = 11

is denoted by H (d). I t is well known th a t H (d) is a fin ite abelian group w ith

respect to Gaussian composition, see for example [9]. The group H (d) is

called the form class group. The order of H (d) is the class number h(d). We

w rite [a, 6, c] for the class containing the form ax2 + bxy + cy2. The iden tity

of H (d ) is the class

1 =

1,0, -d

1, 1,1 - d

i f d = 0 (mod 4),

, i f d = 1 (mod 4).

The inverse o f the class K — [a, 6, c] E H (d) is the class K 1

H (d). Let A i , . . . A s be generators of H (d) chosen so th a t

[a, - b, c] €

h i/i2 ■■•hs = h(d), 1 < h i | h2 \ • • • | hs, o rd ^ * ) = h i( i = 1 , . . . , s).

For K = A * 1 • • • A kss E i/(d ) and L = A [x ■ ■ ■ Ae/ e H (d) we define

X ( K , L ) = e2ni( ^ + - +h^ ) .

I f p is a prim e w ith ^ = 1, we le t x \ and rc2 be the two solutions of x2 = d

(mod 4p), 0 < x < 2p, w ith x \ < x2. We define the class K p E H {d) by

x \ — d

so tha t

K p = P> x i ’ 4p

’ 4p

For K ( j t I ) E -ff(d) the Bernays constant [7, Teil I, §3, §4, pp. 36-68] is

defined by

pa h '

I t is known th a t j (K ,d ) is a nonzero real number such th a t j (K ,d ) =

j ( K _1, d) [26, Lemma 7.6]. For K E H {d) and n E N we define H x{n )

to be the number of integers h satisfying

h2 - d0 < h < 2 n, h2 = d (mod 4n),

Further for n 6 N and K E H (d) we set

n, h,4 n

= K .

YK ( n ) = Y , x ( K , L ) H L (n).L£H{d)

For K E H (d) and a prim e p we set

A (K ,d ,P ) = Y ^ 1 -d=o F

Then we define for K E H(d)

l ( K , d ) = n ( l + x(-K ' K p ) ) l [ A ( K , d , p ) .p > pi?

r f/

We also set

*«> - n (* - £ ) •(f)=i

Finally for K E H(d) we define

TTi/jy- _ * V \ d \ \ ' / r T S - \— 1 ^ l ( ^ ) P ( T J \

48W r ^ * {L ’ K ) J iL id jL±I

We are now ready to state the result o f Muzaffar and W illiam s [26, Theorem

2]. I t is convenient to w rite fo(z) for f(z ) . The power of 2 occurring in the

nonzero rational number r is denoted by v2 (r), so th a t

j,2(24) = u2 (23 • 3) = 3,

1/2 ( t ) = U2^ 2 ' 3-1 ’ =2’

= ^ ( 2' 3 ■ 3-1 ■ 7) = - 3-

CHAPTER 3. KRONECKER’S THEOREM

T heorem 3.4.1. Let K = [a,b,c] € H {d). Set

q0 = a + b + c, q\ = c, q2 = a,

1, i fq i = 2 (mod 4),

1, i f q% = 0 (mod 4), b = 1 (mod 2)

1/2, i f q% = 0 (mod 4), b = 0 (mod 2)

2, i f qi = l (mod 2),

fo r i = 0 ,1,2,

M 0 =

M x =

M 2 =

2a Aq, Ao(2a + 6), ~ ( a + 6 + c)

2aAi, Ai6, y c

A2a, A26, 2A2c

€ # (A^d),

m, = 2 - 21 _ " 2 ( A i ) = <

0, * / Ai = 1,

1, i f Xi = 2,

-2 , i f Xi = 1/2,

/o r i = 0,1,2. Then

b + \ /d/*

fo r i — 0,1,2.

2a =(i:1/4

Prom Theorems 3.3.2 and 3.4.1 we obtain the following theorem.


T heo rem 3.4.2. With the notation o f Theorem 3.4.1,

(i) t am? + bmn + cn2771,71=—OO

^ {i log g ) +mi (gi|))2- ,( /) I o g 2 + E ( K , $ _ E ( M u A?(j)

uu

(M) E ( - i ) "am2 + bmn + cn?

771,71=—OO

TPf {1los © + to*2+ A 2 d )

on) E00 / . \ m -|_n(-1 )*

am2 + bmn + cn2771,71=—OO

. 1,0g g ) +mo(g||))2-W/,log2 + i5(if,d) _ £(Mo,^)

We conclude th is thesis w ith an example illus tra ting Theorem 3.4.2. We

take

a = 1, 6 = 0, c = 19

so tha t

d = -7 6 , / = 2, A = -1 9 , g ) = -1 , M f ) = 1.

K = [1,0,19] e H ( - 7 6 ) ,

qQ = 20, qi = 19, q2 = 1,

A° = - , Ai = 2, A2 = 2,

Mo = [1,1,5] € # ( - 1 9 ) ,



m 0 = 2 — 21“ V2(Ao) = 2 - 4 = -2 .

Muzaffar and W illiam s [26] have shown tha t

E (K , d) = log ( J jL ) , £ (M „, -1 9 ) = 0,

where 0 is the unique real root of x3—2x—2 = 0. Thus, by Theorem 3.4.2(iii),

we have

(_ i)m +n 8tt / l . , - 2 1 . _ . ( 6 \ \

^ m 2 + 1 9 n 2 “ i/7 6 \4 2 ' 3 ' 8 S + g (,21/3 J J47T ( \ 1 1 \

= - 7 f 5 ( 2 lo« 2 - 6 l0s2 + l0 g 9 - 3 1Og2)

m ,n =—oo

47Tlog 0.

We have proved

T h eo rem 3.4.3.

V - ' ( - l ) m+n 47T , „^ m 2 + 19n2m ,n = —oo

S im ilarly we can show from Theorem 3.4.2(i), tha t

V ' 4 ^ = — ^= lo g (2 7 + 5\/29),^ m 2 + 58n2 y/58m ,n = —oo v

a result given by Zucker and Robertson in [38].

We finish by remarking th a t the series in Theorem 3.4.2 are related to

series o f the form V ------------= ------, k e N , see [35], [36], [37].n sinh(\/& 7m) 1 J 1 J


Conclusion

We conclude th is thesis by mentioning a possible line of fu ture research. We

have noted th a t certain in fin ite series evaluations such as

f ; ± v r _ _ j i _ , o g 2 _ ^ L log('2 + V 3 ) i^ m 2 + 3n2 3\/3 2 \/3 V /m ,n = —oo v v

(m,n)^(0,0)

mUL2 = _3 ^ l0g2+2 ^ l0g(2 + V )’m ,n = —oo v v(m ,n )^ (0 ,0 )

“ ( _ l ) m + n ^ 4?T

^ m 2 + 3n2 “ 3 ^3m ,n=—oo v(m ,n )# (0 ,0 )

see Theorem 3.3.7, follow from Kronecker’s lim it form ula but do not appear to

be capable of deduction from Rademacher’s theorem. I t is therefore natural

to conjecture th a t there is a generalization of Rademacher’s theorem which

gives the above results as special cases. Much remains to be investigated.

109


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