Post on 14-Jun-2019
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson-Newton formulas and Dirichlet series
Vicente Muñoz (UCM)
27 de noviembre de 2012
Universidad Carlos III de Madrid
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson formulas and Dirichlet series
1 Classical Poisson formula
2 Dirichlet series
3 Poisson formulas for Dirichlet series
4 Proof of Theorem
5 Further results
(Joint work with Ricardo Pérez-Marco.)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Classical Poisson formula
The Poisson formula reads∑n∈Z
e2πint =∑k∈Z
δk
The waves with frequences λn = n,n = 0,1,2, . . . are resonant at theintegers k ∈ Z.
The fundalmental frequency isλ1 = 1.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Classical Poisson formula
The Poisson formula reads∑n∈Z
e2πint =∑k∈Z
δk
The waves with frequences λn = n,n = 0,1,2, . . . are resonant at theintegers k ∈ Z.
The fundalmental frequency isλ1 = 1.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Classical Poisson formula
The Poisson formula reads∑n∈Z
e2πint =∑k∈Z
δk
The waves with frequences λn = n,n = 0,1,2, . . . are resonant at theintegers k ∈ Z.
The fundalmental frequency isλ1 = 1.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Dirac delta
The Dirac delta is defined as:
δa =
{0, x 6= a∞, x = a
where ∫Rδa = 1
This is not a function.
So what is this?
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Dirac delta
The Dirac delta is defined as:
δa =
{0, x 6= a∞, x = a
where ∫Rδa = 1
This is not a function.
So what is this?
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Dirac delta
The Dirac delta is defined as:
δa =
{0, x 6= a∞, x = a
where ∫Rδa = 1
This is not a function.
So what is this?
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Distributions
We interpret a function as a functional:
〈f ,g〉 =
∫ ∞−∞
f (x)g(x)dx
So the Dirac delta is:
〈δa,g〉 = g(a)
The distributions are (continuous) functionals on our space offunctions.We use exponentially decaying test functions: |g(x)| ≤ Ce−κ|x |
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Distributions
We interpret a function as a functional:
〈f ,g〉 =
∫ ∞−∞
f (x)g(x)dx
So the Dirac delta is:
〈δa,g〉 = g(a)
The distributions are (continuous) functionals on our space offunctions.We use exponentially decaying test functions: |g(x)| ≤ Ce−κ|x |
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Distributions
We interpret a function as a functional:
〈f ,g〉 =
∫ ∞−∞
f (x)g(x)dx
So the Dirac delta is:
〈δa,g〉 = g(a)
The distributions are (continuous) functionals on our space offunctions.
We use exponentially decaying test functions: |g(x)| ≤ Ce−κ|x |
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Distributions
We interpret a function as a functional:
〈f ,g〉 =
∫ ∞−∞
f (x)g(x)dx
So the Dirac delta is:
〈δa,g〉 = g(a)
The distributions are (continuous) functionals on our space offunctions.We use exponentially decaying test functions: |g(x)| ≤ Ce−κ|x |
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Fourier transform
The Fourier transform
f (x) = e2πiαx f (t) = δα
A “basis” of the “space” of functions is given by{e2πiαx ; α ∈ R}, and another one is {δα ; α ∈ R}So
f (x) =∑α∈R
aαe2πiαx ∑α∈R
aαδα
Rewriting, we get the inverse Fourier transform
a(x) =
∫α∈R
a(α)e2πiαxdα ! a(α)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Fourier transform
The Fourier transform
f (x) = e2πiαx f (t) = δα
A “basis” of the “space” of functions is given by{e2πiαx ; α ∈ R},
and another one is {δα ; α ∈ R}So
f (x) =∑α∈R
aαe2πiαx ∑α∈R
aαδα
Rewriting, we get the inverse Fourier transform
a(x) =
∫α∈R
a(α)e2πiαxdα ! a(α)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Fourier transform
The Fourier transform
f (x) = e2πiαx f (t) = δα
A “basis” of the “space” of functions is given by{e2πiαx ; α ∈ R}, and another one is {δα ; α ∈ R}
Sof (x) =
∑α∈R
aαe2πiαx ∑α∈R
aαδα
Rewriting, we get the inverse Fourier transform
a(x) =
∫α∈R
a(α)e2πiαxdα ! a(α)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Fourier transform
The Fourier transform
f (x) = e2πiαx f (t) = δα
A “basis” of the “space” of functions is given by{e2πiαx ; α ∈ R}, and another one is {δα ; α ∈ R}So
f (x) =∑α∈R
aαe2πiαx ∑α∈R
aαδα
Rewriting, we get the inverse Fourier transform
a(x) =
∫α∈R
a(α)e2πiαxdα ! a(α)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Fourier transform
The Fourier transform
f (x) = e2πiαx f (t) = δα
A “basis” of the “space” of functions is given by{e2πiαx ; α ∈ R}, and another one is {δα ; α ∈ R}So
f (x) =∑α∈R
aαe2πiαx ∑α∈R
aαδα
Rewriting, we get the inverse Fourier transform
a(x) =
∫α∈R
a(α)e2πiαxdα ! a(α)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson formula
The Poisson formula∑n∈Z
e2πint =∑k∈Z
δk is now rewritten as
∑n∈Z
∫ ∞−∞
e 2πinxg(x)dx =∑k∈Z
g(k)
for any test function g.
The alternative form∑n∈Z
g(n) =∑k∈Z
g(k)
is the original form of the Poissonformula.
Poisson (1781-1840)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson formula
The Poisson formula∑n∈Z
e2πint =∑k∈Z
δk is now rewritten as
∑n∈Z
∫ ∞−∞
e 2πinxg(x)dx =∑k∈Z
g(k)
for any test function g.
The alternative form∑n∈Z
g(n) =∑k∈Z
g(k)
is the original form of the Poissonformula.
Poisson (1781-1840)V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Reinterpreting the classical Poisson formula
Let λ > 0 be the fundamental frequency.
Then the Poisson formula reads∑n∈Z
e2πiλ
nt = λ∑k∈Z
δλk
This is associated to the Dirichlet series f (s) = 1− e−λs
Left-hand-side:∑
eρt , over zeroes ρ = −2πiλ n of f .
Right-hand-side: Dirac deltas at multiples of fundamentalfrequency of f .
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Reinterpreting the classical Poisson formula
Let λ > 0 be the fundamental frequency.Then the Poisson formula reads∑
n∈Ze
2πiλ
nt = λ∑k∈Z
δλk
This is associated to the Dirichlet series f (s) = 1− e−λs
Left-hand-side:∑
eρt , over zeroes ρ = −2πiλ n of f .
Right-hand-side: Dirac deltas at multiples of fundamentalfrequency of f .
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Reinterpreting the classical Poisson formula
Let λ > 0 be the fundamental frequency.Then the Poisson formula reads∑
n∈Ze
2πiλ
nt = λ∑k∈Z
δλk
This is associated to the Dirichlet series f (s) = 1− e−λs
Left-hand-side:∑
eρt , over zeroes ρ = −2πiλ n of f .
Right-hand-side: Dirac deltas at multiples of fundamentalfrequency of f .
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Reinterpreting the classical Poisson formula
Let λ > 0 be the fundamental frequency.Then the Poisson formula reads∑
n∈Ze
2πiλ
nt = λ∑k∈Z
δλk
This is associated to the Dirichlet series f (s) = 1− e−λs
Left-hand-side:∑
eρt , over zeroes ρ = −2πiλ n of f .
Right-hand-side: Dirac deltas at multiples of fundamentalfrequency of f .
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Reinterpreting the classical Poisson formula
Let λ > 0 be the fundamental frequency.Then the Poisson formula reads∑
n∈Ze
2πiλ
nt = λ∑k∈Z
δλk
This is associated to the Dirichlet series f (s) = 1− e−λs
Left-hand-side:∑
eρt , over zeroes ρ = −2πiλ n of f .
Right-hand-side: Dirac deltas at multiples of fundamentalfrequency of f .
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Dirichlet series
Definition
A Dirichlet series is f (s) = 1 +∑
ane−λns, 0 < λ1 < λ2 < . . .,an ∈ C, and there exists some σ ∈ R such that∑|an|e−λnσ <∞.
Dirichlet (1805-1859)
We shall assume that the Dirichletseries has a meromorphic extensionto the whole of C.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Dirichlet series
Definition
A Dirichlet series is f (s) = 1 +∑
ane−λns, 0 < λ1 < λ2 < . . .,an ∈ C, and there exists some σ ∈ R such that∑|an|e−λnσ <∞.
Dirichlet (1805-1859)
We shall assume that the Dirichletseries has a meromorphic extensionto the whole of C.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Dirichlet series
Definition
A Dirichlet series is f (s) = 1 +∑
ane−λns, 0 < λ1 < λ2 < . . .,an ∈ C, and there exists some σ ∈ R such that∑|an|e−λnσ <∞.
Dirichlet (1805-1859)
We shall assume that the Dirichletseries has a meromorphic extensionto the whole of C.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann zeta function
Riemann zeta function
ζ(s) =∞∑
n=1
1ns = 1 +
∑n≥2
e−(log n)s
Riemann (1826-1866)
Riemann (1859) usedζ(s) to study the num-ber of primes π(x) upto x ∈ R.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann zeta function
The Riemann zeta function hasA single pole at s = 1.
Zeroes at s = −2,−4,−6, . . . (trivial zeroes).Other zeroes with 0 < <s < 1 (the critical strip).
ξ(s) = 12π−s/2s(s − 1)Γ(s/2)ζ(s) satisfies
the functional equation
ξ(1− s) = ξ(s).
So the zeroes are symmetric with respectto <s = 1/2.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann zeta function
The Riemann zeta function hasA single pole at s = 1.Zeroes at s = −2,−4,−6, . . . (trivial zeroes).
Other zeroes with 0 < <s < 1 (the critical strip).
ξ(s) = 12π−s/2s(s − 1)Γ(s/2)ζ(s) satisfies
the functional equation
ξ(1− s) = ξ(s).
So the zeroes are symmetric with respectto <s = 1/2.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann zeta function
The Riemann zeta function hasA single pole at s = 1.Zeroes at s = −2,−4,−6, . . . (trivial zeroes).Other zeroes with 0 < <s < 1 (the critical strip).
ξ(s) = 12π−s/2s(s − 1)Γ(s/2)ζ(s) satisfies
the functional equation
ξ(1− s) = ξ(s).
So the zeroes are symmetric with respectto <s = 1/2.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann zeta function
The Riemann zeta function hasA single pole at s = 1.Zeroes at s = −2,−4,−6, . . . (trivial zeroes).Other zeroes with 0 < <s < 1 (the critical strip).
ξ(s) = 12π−s/2s(s − 1)Γ(s/2)ζ(s) satisfies
the functional equation
ξ(1− s) = ξ(s).
So the zeroes are symmetric with respectto <s = 1/2.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann zeta function
The Riemann zeta function hasA single pole at s = 1.Zeroes at s = −2,−4,−6, . . . (trivial zeroes).Other zeroes with 0 < <s < 1 (the critical strip).
ξ(s) = 12π−s/2s(s − 1)Γ(s/2)ζ(s) satisfies
the functional equation
ξ(1− s) = ξ(s).
So the zeroes are symmetric with respectto <s = 1/2.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann zeta function
The Riemann zeta function hasA single pole at s = 1.Zeroes at s = −2,−4,−6, . . . (trivial zeroes).Other zeroes with 0 < <s < 1 (the critical strip).
ξ(s) = 12π−s/2s(s − 1)Γ(s/2)ζ(s) satisfies
the functional equation
ξ(1− s) = ξ(s).
So the zeroes are symmetric with respectto <s = 1/2.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann hypothesis
Riemann hypothesis
All non-trivial zeroes of ζ satisfy <s = 1/2.
First zeroes: ρk = 1/2 + iγk , where γk =14,13472521,02204025,01085830,42487632,935062, . . .
One of the most important open problems in mathematics!The oldest one in the list of the Millenium problems.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann hypothesis
Riemann hypothesis
All non-trivial zeroes of ζ satisfy <s = 1/2.
First zeroes: ρk = 1/2 + iγk , where γk =14,13472521,02204025,01085830,42487632,935062, . . .
One of the most important open problems in mathematics!The oldest one in the list of the Millenium problems.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann hypothesis
Riemann hypothesis
All non-trivial zeroes of ζ satisfy <s = 1/2.
First zeroes: ρk = 1/2 + iγk , where γk =14,13472521,02204025,01085830,42487632,935062, . . .
One of the most important open problems in mathematics!The oldest one in the list of the Millenium problems.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Riemann hypothesis
Riemann hypothesis
All non-trivial zeroes of ζ satisfy <s = 1/2.
First zeroes: ρk = 1/2 + iγk , where γk =14,13472521,02204025,01085830,42487632,935062, . . .
One of the most important open problems in mathematics!The oldest one in the list of the Millenium problems.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson formula for a Dirichlet series
Let f (s) = 1 +∑
ane−λns.
Write
− log f (s) =∑
b~r e−(λ1r1+...+λ`r`)s =
∑b~r e−〈~λ,~r 〉s
Theorem (V. M. & R. Pérez-Marco)
Let D =∑
nρρ be the divisor of zeroes/poles of f . Then, asdistributions, ∑
nρeρt =∑〈~λ,~r 〉b~r δ〈~λ,~r 〉
on R+
W (t) =∑
nρeρt is called the Newton-Cramer distribution.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson formula for a Dirichlet series
Let f (s) = 1 +∑
ane−λns. Write
− log f (s) =∑
b~r e−(λ1r1+...+λ`r`)s
=∑
b~r e−〈~λ,~r 〉s
Theorem (V. M. & R. Pérez-Marco)
Let D =∑
nρρ be the divisor of zeroes/poles of f . Then, asdistributions, ∑
nρeρt =∑〈~λ,~r 〉b~r δ〈~λ,~r 〉
on R+
W (t) =∑
nρeρt is called the Newton-Cramer distribution.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson formula for a Dirichlet series
Let f (s) = 1 +∑
ane−λns. Write
− log f (s) =∑
b~r e−(λ1r1+...+λ`r`)s =
∑b~r e−〈~λ,~r 〉s
Theorem (V. M. & R. Pérez-Marco)
Let D =∑
nρρ be the divisor of zeroes/poles of f . Then, asdistributions, ∑
nρeρt =∑〈~λ,~r 〉b~r δ〈~λ,~r 〉
on R+
W (t) =∑
nρeρt is called the Newton-Cramer distribution.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson formula for a Dirichlet series
Let f (s) = 1 +∑
ane−λns. Write
− log f (s) =∑
b~r e−(λ1r1+...+λ`r`)s =
∑b~r e−〈~λ,~r 〉s
Theorem (V. M. & R. Pérez-Marco)
Let D =∑
nρρ be the divisor of zeroes/poles of f . Then, asdistributions, ∑
nρeρt =∑〈~λ,~r 〉b~r δ〈~λ,~r 〉
on R+
W (t) =∑
nρeρt is called the Newton-Cramer distribution.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson formula for a Dirichlet series
Let f (s) = 1 +∑
ane−λns. Write
− log f (s) =∑
b~r e−(λ1r1+...+λ`r`)s =
∑b~r e−〈~λ,~r 〉s
Theorem (V. M. & R. Pérez-Marco)
Let D =∑
nρρ be the divisor of zeroes/poles of f . Then, asdistributions, ∑
nρeρt =∑〈~λ,~r 〉b~r δ〈~λ,~r 〉
on R+
W (t) =∑
nρeρt is called the Newton-Cramer distribution.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Poisson formula for a Dirichlet series
Let f (s) = 1 +∑
ane−λns. Write
− log f (s) =∑
b~r e−(λ1r1+...+λ`r`)s =
∑b~r e−〈~λ,~r 〉s
Theorem (V. M. & R. Pérez-Marco)
Let D =∑
nρρ be the divisor of zeroes/poles of f . Then, asdistributions, ∑
nρeρt =∑〈~λ,~r 〉b~r δ〈~λ,~r 〉
on R+
W (t) =∑
nρeρt is called the Newton-Cramer distribution.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Example: the zeta function
ζ(s) =∑n≥1
1ns
=∏
p
(1 +
1ps +
1p2s + . . .
)=∏
p
(1− 1
ps
)−1
− log ζ(s) =∑
p
log(1− p−s)
=∑
p
∑k≥1
−1k
p−ks
=∑p,k
−1k
e−(k log p)s
So∑nρeρt = −
∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Example: the zeta function
ζ(s) =∑n≥1
1ns =
∏p
(1 +
1ps +
1p2s + . . .
)
=∏
p
(1− 1
ps
)−1
− log ζ(s) =∑
p
log(1− p−s)
=∑
p
∑k≥1
−1k
p−ks
=∑p,k
−1k
e−(k log p)s
So∑nρeρt = −
∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Example: the zeta function
ζ(s) =∑n≥1
1ns =
∏p
(1 +
1ps +
1p2s + . . .
)=∏
p
(1− 1
ps
)−1
− log ζ(s) =∑
p
log(1− p−s)
=∑
p
∑k≥1
−1k
p−ks
=∑p,k
−1k
e−(k log p)s
So∑nρeρt = −
∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Example: the zeta function
ζ(s) =∑n≥1
1ns =
∏p
(1 +
1ps +
1p2s + . . .
)=∏
p
(1− 1
ps
)−1
− log ζ(s) =∑
p
log(1− p−s)
=∑
p
∑k≥1
−1k
p−ks
=∑p,k
−1k
e−(k log p)s
So∑nρeρt = −
∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Example: the zeta function
ζ(s) =∑n≥1
1ns =
∏p
(1 +
1ps +
1p2s + . . .
)=∏
p
(1− 1
ps
)−1
− log ζ(s) =∑
p
log(1− p−s)
=∑
p
∑k≥1
−1k
p−ks
=∑p,k
−1k
e−(k log p)s
So∑nρeρt = −
∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Example: the zeta function
ζ(s) =∑n≥1
1ns =
∏p
(1 +
1ps +
1p2s + . . .
)=∏
p
(1− 1
ps
)−1
− log ζ(s) =∑
p
log(1− p−s)
=∑
p
∑k≥1
−1k
p−ks
=∑p,k
−1k
e−(k log p)s
So∑nρeρt = −
∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Example: the zeta function
ζ(s) =∑n≥1
1ns =
∏p
(1 +
1ps +
1p2s + . . .
)=∏
p
(1− 1
ps
)−1
− log ζ(s) =∑
p
log(1− p−s)
=∑
p
∑k≥1
−1k
p−ks
=∑p,k
−1k
e−(k log p)s
So∑nρeρt = −
∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Cramer function
Write ρ = 12 + iγ, for the non-trivial zeroes.
Let W0(t) = −et +∑n≥1
e−2nt = −et +e−2t
1− e−2t
et/2∑
eiγt + W0(t) = −∑p,k
(log p)δk log p
The Cramer function is
V (t) =∑γ>0
eiγt
Then
et/2V (t) + et/2V (−t) + W0(t) = −∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Cramer function
Write ρ = 12 + iγ, for the non-trivial zeroes.
Let W0(t) = −et +∑n≥1
e−2nt = −et +e−2t
1− e−2t
et/2∑
eiγt + W0(t) = −∑p,k
(log p)δk log p
The Cramer function is
V (t) =∑γ>0
eiγt
Then
et/2V (t) + et/2V (−t) + W0(t) = −∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Cramer function
Write ρ = 12 + iγ, for the non-trivial zeroes.
Let W0(t) = −et +∑n≥1
e−2nt = −et +e−2t
1− e−2t
et/2∑
eiγt + W0(t) = −∑p,k
(log p)δk log p
The Cramer function is
V (t) =∑γ>0
eiγt
Then
et/2V (t) + et/2V (−t) + W0(t) = −∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Cramer function
Write ρ = 12 + iγ, for the non-trivial zeroes.
Let W0(t) = −et +∑n≥1
e−2nt = −et +e−2t
1− e−2t
et/2∑
eiγt + W0(t) = −∑p,k
(log p)δk log p
The Cramer function is
V (t) =∑γ>0
eiγt
Then
et/2V (t) + et/2V (−t) + W0(t) = −∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Cramer function
Write ρ = 12 + iγ, for the non-trivial zeroes.
Let W0(t) = −et +∑n≥1
e−2nt = −et +e−2t
1− e−2t
et/2∑
eiγt + W0(t) = −∑p,k
(log p)δk log p
The Cramer function is
V (t) =∑γ>0
eiγt
Then
et/2V (t) + et/2V (−t) + W0(t) = −∑p,k
(log p)δk log p
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Newton formulas
Let P(z) = zn + a1zn−1 + . . .+ an be a polynomial, and letα1, . . . , αn be its zeroes.
Set z = es. We get a Dirichlet series
f (s) = e−nsP(es) = 1 + a1e−s + . . .+ ane−ns
A zero of f is of the form
eρjk = αje2πik
where j = 1, . . . ,n, k ∈ Z.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Newton formulas
Let P(z) = zn + a1zn−1 + . . .+ an be a polynomial, and letα1, . . . , αn be its zeroes.Set z = es. We get a Dirichlet series
f (s) = e−nsP(es) = 1 + a1e−s + . . .+ ane−ns
A zero of f is of the form
eρjk = αje2πik
where j = 1, . . . ,n, k ∈ Z.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Newton formulas
Let P(z) = zn + a1zn−1 + . . .+ an be a polynomial, and letα1, . . . , αn be its zeroes.Set z = es. We get a Dirichlet series
f (s) = e−nsP(es) = 1 + a1e−s + . . .+ ane−ns
A zero of f is of the form
eρjk = αje2πik
where j = 1, . . . ,n, k ∈ Z.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Newton formulas
The Newton-Cramer distribution is
W (f ) =∑
eρt =∑j,k
αtj e
2πikt
=∑
j
αtj
∑k
e2πikt
=∑
j
αtj
∑m
δm (by the Poisson formula)
=∑
m
Smδm (where Sm =∑
j αmj are the Newton sums)
W (f )(t) generalizes the Newton sums to exponents t ∈ R+,and they are non-zero only for t = m ∈ Z+.
Poisson-Newton formula ⇐⇒ Newton relations for P(z).
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Newton formulas
The Newton-Cramer distribution is
W (f ) =∑
eρt =∑j,k
αtj e
2πikt
=∑
j
αtj
∑k
e2πikt
=∑
j
αtj
∑m
δm (by the Poisson formula)
=∑
m
Smδm (where Sm =∑
j αmj are the Newton sums)
W (f )(t) generalizes the Newton sums to exponents t ∈ R+,and they are non-zero only for t = m ∈ Z+.
Poisson-Newton formula ⇐⇒ Newton relations for P(z).
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Newton formulas
The Newton-Cramer distribution is
W (f ) =∑
eρt =∑j,k
αtj e
2πikt
=∑
j
αtj
∑k
e2πikt
=∑
j
αtj
∑m
δm (by the Poisson formula)
=∑
m
Smδm (where Sm =∑
j αmj are the Newton sums)
W (f )(t) generalizes the Newton sums to exponents t ∈ R+,and they are non-zero only for t = m ∈ Z+.
Poisson-Newton formula ⇐⇒ Newton relations for P(z).
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Newton formulas
The Newton-Cramer distribution is
W (f ) =∑
eρt =∑j,k
αtj e
2πikt
=∑
j
αtj
∑k
e2πikt
=∑
j
αtj
∑m
δm (by the Poisson formula)
=∑
m
Smδm (where Sm =∑
j αmj are the Newton sums)
W (f )(t) generalizes the Newton sums to exponents t ∈ R+,and they are non-zero only for t = m ∈ Z+.
Poisson-Newton formula ⇐⇒ Newton relations for P(z).
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Newton formulas
The Newton-Cramer distribution is
W (f ) =∑
eρt =∑j,k
αtj e
2πikt
=∑
j
αtj
∑k
e2πikt
=∑
j
αtj
∑m
δm (by the Poisson formula)
=∑
m
Smδm (where Sm =∑
j αmj are the Newton sums)
W (f )(t) generalizes the Newton sums to exponents t ∈ R+,and they are non-zero only for t = m ∈ Z+.
Poisson-Newton formula ⇐⇒ Newton relations for P(z).
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Newton formulas
The Newton-Cramer distribution is
W (f ) =∑
eρt =∑j,k
αtj e
2πikt
=∑
j
αtj
∑k
e2πikt
=∑
j
αtj
∑m
δm (by the Poisson formula)
=∑
m
Smδm (where Sm =∑
j αmj are the Newton sums)
W (f )(t) generalizes the Newton sums to exponents t ∈ R+,and they are non-zero only for t = m ∈ Z+.
Poisson-Newton formula ⇐⇒ Newton relations for P(z).
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Defining the Newton-Cramer distribution
The exponent of convergence is the minimum integer d ≥ 0such that
∑|nρ| |ρ|−d <∞.
Fix σ ∈ C and let
Kd (t) =∑ nρ
(ρ− σ)d (e(ρ−σ)t − 1)1R+
which is a continuous function. Define
W (f ) = eσt Dd
Dtd Kd (t)
W (f ) does not depend on σ over R+.There is a contribution at zero.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Defining the Newton-Cramer distribution
The exponent of convergence is the minimum integer d ≥ 0such that
∑|nρ| |ρ|−d <∞. Fix σ ∈ C and let
Kd (t) =∑ nρ
(ρ− σ)d (e(ρ−σ)t − 1)1R+
which is a continuous function.
Define
W (f ) = eσt Dd
Dtd Kd (t)
W (f ) does not depend on σ over R+.There is a contribution at zero.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Defining the Newton-Cramer distribution
The exponent of convergence is the minimum integer d ≥ 0such that
∑|nρ| |ρ|−d <∞. Fix σ ∈ C and let
Kd (t) =∑ nρ
(ρ− σ)d (e(ρ−σ)t − 1)1R+
which is a continuous function. Define
W (f ) = eσt Dd
Dtd Kd (t)
W (f ) does not depend on σ over R+.There is a contribution at zero.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Defining the Newton-Cramer distribution
The exponent of convergence is the minimum integer d ≥ 0such that
∑|nρ| |ρ|−d <∞. Fix σ ∈ C and let
Kd (t) =∑ nρ
(ρ− σ)d (e(ρ−σ)t − 1)1R+
which is a continuous function. Define
W (f ) = eσt Dd
Dtd Kd (t)
W (f ) does not depend on σ over R+.
There is a contribution at zero.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Defining the Newton-Cramer distribution
The exponent of convergence is the minimum integer d ≥ 0such that
∑|nρ| |ρ|−d <∞. Fix σ ∈ C and let
Kd (t) =∑ nρ
(ρ− σ)d (e(ρ−σ)t − 1)1R+
which is a continuous function. Define
W (f ) = eσt Dd
Dtd Kd (t)
W (f ) does not depend on σ over R+.There is a contribution at zero.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Hadamard interpolation
Let Em(z) = (1− z)ez+ 12 z2+... 1
m zmbe the Weierstrass factor.
Hadamard interpolation says that
f (s) = eQf (s)∏ρ
Ed−1
(s − σρ− σ
)nρ
The genus of f is g = m«ax{deg Qf ,d − 1}.Let
G(s) =∑ρ
nρ
(1
ρ− s−
d−2∑`=0
(s − σ)`
(ρ− σ)`+1
)Then
f ′
f= (log f )′ = Q′f + G
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Hadamard interpolation
Let Em(z) = (1− z)ez+ 12 z2+... 1
m zmbe the Weierstrass factor.
Hadamard interpolation says that
f (s) = eQf (s)∏ρ
Ed−1
(s − σρ− σ
)nρ
The genus of f is g = m«ax{deg Qf ,d − 1}.Let
G(s) =∑ρ
nρ
(1
ρ− s−
d−2∑`=0
(s − σ)`
(ρ− σ)`+1
)Then
f ′
f= (log f )′ = Q′f + G
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Hadamard interpolation
Let Em(z) = (1− z)ez+ 12 z2+... 1
m zmbe the Weierstrass factor.
Hadamard interpolation says that
f (s) = eQf (s)∏ρ
Ed−1
(s − σρ− σ
)nρ
The genus of f is g = m«ax{deg Qf ,d − 1}.
Let
G(s) =∑ρ
nρ
(1
ρ− s−
d−2∑`=0
(s − σ)`
(ρ− σ)`+1
)Then
f ′
f= (log f )′ = Q′f + G
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Hadamard interpolation
Let Em(z) = (1− z)ez+ 12 z2+... 1
m zmbe the Weierstrass factor.
Hadamard interpolation says that
f (s) = eQf (s)∏ρ
Ed−1
(s − σρ− σ
)nρ
The genus of f is g = m«ax{deg Qf ,d − 1}.Let
G(s) =∑ρ
nρ
(1
ρ− s−
d−2∑`=0
(s − σ)`
(ρ− σ)`+1
)
Thenf ′
f= (log f )′ = Q′f + G
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Hadamard interpolation
Let Em(z) = (1− z)ez+ 12 z2+... 1
m zmbe the Weierstrass factor.
Hadamard interpolation says that
f (s) = eQf (s)∏ρ
Ed−1
(s − σρ− σ
)nρ
The genus of f is g = m«ax{deg Qf ,d − 1}.Let
G(s) =∑ρ
nρ
(1
ρ− s−
d−2∑`=0
(s − σ)`
(ρ− σ)`+1
)Then
f ′
f= (log f )′ = Q′f + G
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Laplace transform of W (f )
L(W (f )) = 〈W (f ),e−st〉
=
⟨Dd
Dtd Kd (t),e(σ−s)t⟩
=
∫ ∞0
(−1)d∑ρ
nρ1
(ρ− σ)d (e(ρ−σ)t − 1)dd
dtd e(σ−s)tdt
=∑ρ
nρ(s − σ)d
(ρ− σ)d
∫ ∞0
(e(ρ−s)t − e(σ−s)t )dt
= −∑ρ
nρ(s − σ)d
(ρ− σ)d
(1
ρ− s− 1σ − s
)= G(s)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Laplace transform of W (f )
L(W (f )) = 〈W (f ),e−st〉
=
⟨Dd
Dtd Kd (t),e(σ−s)t⟩
=
∫ ∞0
(−1)d∑ρ
nρ1
(ρ− σ)d (e(ρ−σ)t − 1)dd
dtd e(σ−s)tdt
=∑ρ
nρ(s − σ)d
(ρ− σ)d
∫ ∞0
(e(ρ−s)t − e(σ−s)t )dt
= −∑ρ
nρ(s − σ)d
(ρ− σ)d
(1
ρ− s− 1σ − s
)= G(s)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Laplace transform of W (f )
L(W (f )) = 〈W (f ),e−st〉
=
⟨Dd
Dtd Kd (t),e(σ−s)t⟩
=
∫ ∞0
(−1)d∑ρ
nρ1
(ρ− σ)d (e(ρ−σ)t − 1)dd
dtd e(σ−s)tdt
=∑ρ
nρ(s − σ)d
(ρ− σ)d
∫ ∞0
(e(ρ−s)t − e(σ−s)t )dt
= −∑ρ
nρ(s − σ)d
(ρ− σ)d
(1
ρ− s− 1σ − s
)= G(s)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Laplace transform of W (f )
L(W (f )) = 〈W (f ),e−st〉
=
⟨Dd
Dtd Kd (t),e(σ−s)t⟩
=
∫ ∞0
(−1)d∑ρ
nρ1
(ρ− σ)d (e(ρ−σ)t − 1)dd
dtd e(σ−s)tdt
=∑ρ
nρ(s − σ)d
(ρ− σ)d
∫ ∞0
(e(ρ−s)t − e(σ−s)t )dt
= −∑ρ
nρ(s − σ)d
(ρ− σ)d
(1
ρ− s− 1σ − s
)= G(s)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Laplace transform of W (f )
L(W (f )) = 〈W (f ),e−st〉
=
⟨Dd
Dtd Kd (t),e(σ−s)t⟩
=
∫ ∞0
(−1)d∑ρ
nρ1
(ρ− σ)d (e(ρ−σ)t − 1)dd
dtd e(σ−s)tdt
=∑ρ
nρ(s − σ)d
(ρ− σ)d
∫ ∞0
(e(ρ−s)t − e(σ−s)t )dt
= −∑ρ
nρ(s − σ)d
(ρ− σ)d
(1
ρ− s− 1σ − s
)
= G(s)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Laplace transform of W (f )
L(W (f )) = 〈W (f ),e−st〉
=
⟨Dd
Dtd Kd (t),e(σ−s)t⟩
=
∫ ∞0
(−1)d∑ρ
nρ1
(ρ− σ)d (e(ρ−σ)t − 1)dd
dtd e(σ−s)tdt
=∑ρ
nρ(s − σ)d
(ρ− σ)d
∫ ∞0
(e(ρ−s)t − e(σ−s)t )dt
= −∑ρ
nρ(s − σ)d
(ρ− σ)d
(1
ρ− s− 1σ − s
)= G(s)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Proof of Poisson-Newton formula
Fromf ′
f= (log f )′ = Q′f + G
= (c0 + c1s + . . .+ cg−1sg−1) + G
take inverse Laplace tranform to get
W (f ) = L−1(G) = −g−1∑j=1
cjδj0 + L−1(f ′/f )
For a Dirichlet series f , log f =∑
b~r e−〈~λ,~r 〉s. Hence
L−1(log f ) =∑
b~rδ〈~λ,~r 〉and
L−1((log f )′) = s∑
b~rδ〈~λ,~r 〉 =∑
b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Proof of Poisson-Newton formula
Fromf ′
f= (log f )′ = Q′f + G = (c0 + c1s + . . .+ cg−1sg−1) + G
take inverse Laplace tranform to get
W (f ) = L−1(G) = −g−1∑j=1
cjδj0 + L−1(f ′/f )
For a Dirichlet series f , log f =∑
b~r e−〈~λ,~r 〉s. Hence
L−1(log f ) =∑
b~rδ〈~λ,~r 〉and
L−1((log f )′) = s∑
b~rδ〈~λ,~r 〉 =∑
b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Proof of Poisson-Newton formula
Fromf ′
f= (log f )′ = Q′f + G = (c0 + c1s + . . .+ cg−1sg−1) + G
take inverse Laplace tranform to get
W (f ) = L−1(G)
= −g−1∑j=1
cjδj0 + L−1(f ′/f )
For a Dirichlet series f , log f =∑
b~r e−〈~λ,~r 〉s. Hence
L−1(log f ) =∑
b~rδ〈~λ,~r 〉and
L−1((log f )′) = s∑
b~rδ〈~λ,~r 〉 =∑
b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Proof of Poisson-Newton formula
Fromf ′
f= (log f )′ = Q′f + G = (c0 + c1s + . . .+ cg−1sg−1) + G
take inverse Laplace tranform to get
W (f ) = L−1(G) = −g−1∑j=1
cjδj0 + L−1(f ′/f )
For a Dirichlet series f , log f =∑
b~r e−〈~λ,~r 〉s. Hence
L−1(log f ) =∑
b~rδ〈~λ,~r 〉and
L−1((log f )′) = s∑
b~rδ〈~λ,~r 〉 =∑
b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Proof of Poisson-Newton formula
Fromf ′
f= (log f )′ = Q′f + G = (c0 + c1s + . . .+ cg−1sg−1) + G
take inverse Laplace tranform to get
W (f ) = L−1(G) = −g−1∑j=1
cjδj0 + L−1(f ′/f )
For a Dirichlet series f , log f =∑
b~r e−〈~λ,~r 〉s.
Hence
L−1(log f ) =∑
b~rδ〈~λ,~r 〉and
L−1((log f )′) = s∑
b~rδ〈~λ,~r 〉 =∑
b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Proof of Poisson-Newton formula
Fromf ′
f= (log f )′ = Q′f + G = (c0 + c1s + . . .+ cg−1sg−1) + G
take inverse Laplace tranform to get
W (f ) = L−1(G) = −g−1∑j=1
cjδj0 + L−1(f ′/f )
For a Dirichlet series f , log f =∑
b~r e−〈~λ,~r 〉s. Hence
L−1(log f ) =∑
b~rδ〈~λ,~r 〉
and
L−1((log f )′) = s∑
b~rδ〈~λ,~r 〉 =∑
b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Proof of Poisson-Newton formula
Fromf ′
f= (log f )′ = Q′f + G = (c0 + c1s + . . .+ cg−1sg−1) + G
take inverse Laplace tranform to get
W (f ) = L−1(G) = −g−1∑j=1
cjδj0 + L−1(f ′/f )
For a Dirichlet series f , log f =∑
b~r e−〈~λ,~r 〉s. Hence
L−1(log f ) =∑
b~rδ〈~λ,~r 〉and
L−1((log f )′) = s∑
b~rδ〈~λ,~r 〉 =∑
b~r 〈~λ,~r 〉 δ〈~λ,~r 〉
QED
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Proof of Poisson-Newton formula
Fromf ′
f= (log f )′ = Q′f + G = (c0 + c1s + . . .+ cg−1sg−1) + G
take inverse Laplace tranform to get
W (f ) = L−1(G) = −g−1∑j=1
cjδj0 + L−1(f ′/f )
For a Dirichlet series f , log f =∑
b~r e−〈~λ,~r 〉s. Hence
L−1(log f ) =∑
b~rδ〈~λ,~r 〉and
L−1((log f )′) = s∑
b~rδ〈~λ,~r 〉 =∑
b~r 〈~λ,~r 〉 δ〈~λ,~r 〉 QED
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
On the genus of Dirichlet series
Corollary (V. M. & R. Pérez-Marco)
For a Dirichlet series, we have
d = g + 1d ≥ 2
(Just compare the order of the two distributions in the two sidesof the Poisson-Newton formula)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
On the genus of Dirichlet series
Corollary (V. M. & R. Pérez-Marco)
For a Dirichlet series, we haved = g + 1
d ≥ 2
(Just compare the order of the two distributions in the two sidesof the Poisson-Newton formula)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
On the genus of Dirichlet series
Corollary (V. M. & R. Pérez-Marco)
For a Dirichlet series, we haved = g + 1d ≥ 2
(Just compare the order of the two distributions in the two sidesof the Poisson-Newton formula)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
On the genus of Dirichlet series
Corollary (V. M. & R. Pérez-Marco)
For a Dirichlet series, we haved = g + 1d ≥ 2
(Just compare the order of the two distributions in the two sidesof the Poisson-Newton formula)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
On the genus of Dirichlet series
Corollary (V. M. & R. Pérez-Marco)
For a Dirichlet series, we haved = g + 1d ≥ 2
(Just compare the order of the two distributions in the two sidesof the Poisson-Newton formula)
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Symmetric Poisson-Newton formula
For f (s) = 1 +∑
ane−λns, an ∈ R, we have∑nρeρ|t | = 2
∑c2lδ
(2l)0 +
∑~r∈Λ∪(−Λ)
b|~r |〈~λ, |~r |〉δ〈~λ,~r〉
on R.
For the classical Poisson formula:∑n∈Z
e2πiλ
nt = λδ0 +∑k∈Z∗
λδkλ
We have
c0 = λ/2bk = 1/k , k ∈ Z∗
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Symmetric Poisson-Newton formula
For f (s) = 1 +∑
ane−λns, an ∈ R, we have∑nρeρ|t | = 2
∑c2lδ
(2l)0 +
∑~r∈Λ∪(−Λ)
b|~r |〈~λ, |~r |〉δ〈~λ,~r〉
on R.For the classical Poisson formula:∑
n∈Ze
2πiλ
nt = λδ0 +∑k∈Z∗
λδkλ
We have
c0 = λ/2bk = 1/k , k ∈ Z∗
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Symmetric Poisson-Newton formula
For f (s) = 1 +∑
ane−λns, an ∈ R, we have∑nρeρ|t | = 2
∑c2lδ
(2l)0 +
∑~r∈Λ∪(−Λ)
b|~r |〈~λ, |~r |〉δ〈~λ,~r〉
on R.For the classical Poisson formula:∑
n∈Ze
2πiλ
nt = λδ0 +∑k∈Z∗
λδkλ
We havec0 = λ/2
bk = 1/k , k ∈ Z∗
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Symmetric Poisson-Newton formula
For f (s) = 1 +∑
ane−λns, an ∈ R, we have∑nρeρ|t | = 2
∑c2lδ
(2l)0 +
∑~r∈Λ∪(−Λ)
b|~r |〈~λ, |~r |〉δ〈~λ,~r〉
on R.For the classical Poisson formula:∑
n∈Ze
2πiλ
nt = λδ0 +∑k∈Z∗
λδkλ
We havec0 = λ/2bk = 1/k , k ∈ Z∗
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Explicit formulas
For the Riemman zeta function,∑γ>0
eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑
(δk log p + δ−k log p)
= c0δ0−∑
p−k/2(log p)(δk log p+δ−k log p)
For a test function ϕ, we have∑ϕ(γ)+W0[ϕ] = c0ϕ(0)−
∑p−k/2(log p)(ϕ(k log p)+ϕ(−k log p)),
where W0[ϕ] =∫
e−|t |/2W0(|t |)ϕ(t)dt .
These are known as Explicit formulas.They do not depend on the functional equation.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Explicit formulas
For the Riemman zeta function,∑γ>0
eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑
(δk log p + δ−k log p)
= c0δ0−∑
p−k/2(log p)(δk log p+δ−k log p)
For a test function ϕ, we have∑ϕ(γ)+W0[ϕ] = c0ϕ(0)−
∑p−k/2(log p)(ϕ(k log p)+ϕ(−k log p)),
where W0[ϕ] =∫
e−|t |/2W0(|t |)ϕ(t)dt .
These are known as Explicit formulas.They do not depend on the functional equation.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Explicit formulas
For the Riemman zeta function,∑γ>0
eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑
(δk log p + δ−k log p)
= c0δ0−∑
p−k/2(log p)(δk log p+δ−k log p)
For a test function ϕ, we have∑ϕ(γ)+W0[ϕ] = c0ϕ(0)−
∑p−k/2(log p)(ϕ(k log p)+ϕ(−k log p)),
where W0[ϕ] =∫
e−|t |/2W0(|t |)ϕ(t)dt .
These are known as Explicit formulas.They do not depend on the functional equation.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Explicit formulas
For the Riemman zeta function,∑γ>0
eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑
(δk log p + δ−k log p)
= c0δ0−∑
p−k/2(log p)(δk log p+δ−k log p)
For a test function ϕ, we have∑ϕ(γ)+W0[ϕ] = c0ϕ(0)−
∑p−k/2(log p)(ϕ(k log p)+ϕ(−k log p)),
where W0[ϕ] =∫
e−|t |/2W0(|t |)ϕ(t)dt .
These are known as Explicit formulas.
They do not depend on the functional equation.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Explicit formulas
For the Riemman zeta function,∑γ>0
eiγ|t | + e−|t |/2W0(|t |) = c0δ0 − e−|t |/2∑
(δk log p + δ−k log p)
= c0δ0−∑
p−k/2(log p)(δk log p+δ−k log p)
For a test function ϕ, we have∑ϕ(γ)+W0[ϕ] = c0ϕ(0)−
∑p−k/2(log p)(ϕ(k log p)+ϕ(−k log p)),
where W0[ϕ] =∫
e−|t |/2W0(|t |)ϕ(t)dt .
These are known as Explicit formulas.They do not depend on the functional equation.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Other results
Converse: to a Poisson-Newton formula we associate aDirichlet series.
General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.Selberg Trace formula in geometry.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Other results
Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.
The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.Selberg Trace formula in geometry.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Other results
Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.
Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.Selberg Trace formula in geometry.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Other results
Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.
Explicit formulas in number theory.Selberg Trace formula in geometry.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Other results
Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.
Selberg Trace formula in geometry.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series
Classical Poisson formula Dirichlet series Poisson formulas for Dirichlet series Proof of Theorem Further results
Other results
Converse: to a Poisson-Newton formula we associate aDirichlet series.General Poisson-Newton formula for f meromorphic offinite order.The divisor of a Dirichlet series is not contained in aleft-directed cone.Summation formulas: Abel-Plana, Euler-MacLaurin, etc.Explicit formulas in number theory.Selberg Trace formula in geometry.
V. Muñoz UCM
Poisson-Newton formulas and Dirichlet series