#Rieman Hyp_Prime Numbers & Physics
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Transcript of #Rieman Hyp_Prime Numbers & Physics
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Counting prime numbers, or listening to them
Byungchul Cha
Muhlenberg College
Mount St. Mary’s UniversityACM/MAA Lecture Series
February 9, 2011
1 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28
= 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7
= 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300
= 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163
= 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163
. (163 is a prime number.)2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011
is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021
= 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4
= 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22
= (45 − 2)(45 + 2) = 43 · 47.21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727
is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Warm-up
Let’s try to factor some integers:
28 = 4 · 7 = 22· 7.
16300 = 100 · 163 = 22· 52· 163. (163 is a prime number.)
2011 is a prime number.Must check all primes 2, 3, 5, . . . , 41, 43 ≤
√2011 = 44.8 . . . .
2021 = 2025 − 4 = 452− 22 = (45 − 2)(45 + 2) = 43 · 47.
21727 is a prime number. (What a prime location!)Must check 34 primes: 2, 3, 5, . . . , 137, 139 ≤
√21727.
2 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Bit more challenging
188198812920607963838697239461650439807163563379
417382700763356422988859715234665485319060606504
743045317388011303396716199692321205734031879550
656996221305168759307650257059 =
398075086424064937397125500550386491199064362342
526708406385189575946388957261768583317
*
472772146107435302536223071973048224632914695302
097116459852171130520711256363590397527
This number (188...), whose nickname is RSA-576, wasfactored in 2003 by a team led by J. Franke and T. Kleinjung.The point of these exercises is
Primes numbers seem to behave randomly.
3 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Bit more challenging
188198812920607963838697239461650439807163563379
417382700763356422988859715234665485319060606504
743045317388011303396716199692321205734031879550
656996221305168759307650257059 =
398075086424064937397125500550386491199064362342
526708406385189575946388957261768583317
*
472772146107435302536223071973048224632914695302
097116459852171130520711256363590397527
This number (188...), whose nickname is RSA-576, wasfactored in 2003 by a team led by J. Franke and T. Kleinjung.The point of these exercises is
Primes numbers seem to behave randomly.
3 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Bit more challenging
188198812920607963838697239461650439807163563379
417382700763356422988859715234665485319060606504
743045317388011303396716199692321205734031879550
656996221305168759307650257059 =
398075086424064937397125500550386491199064362342
526708406385189575946388957261768583317
*
472772146107435302536223071973048224632914695302
097116459852171130520711256363590397527
This number (188...), whose nickname is RSA-576, wasfactored in 2003 by a team led by J. Franke and T. Kleinjung.
The point of these exercises is
Primes numbers seem to behave randomly.
3 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Bit more challenging
188198812920607963838697239461650439807163563379
417382700763356422988859715234665485319060606504
743045317388011303396716199692321205734031879550
656996221305168759307650257059 =
398075086424064937397125500550386491199064362342
526708406385189575946388957261768583317
*
472772146107435302536223071973048224632914695302
097116459852171130520711256363590397527
This number (188...), whose nickname is RSA-576, wasfactored in 2003 by a team led by J. Franke and T. Kleinjung.The point of these exercises is
Primes numbers seem to behave randomly.
3 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Moral of this talk
Each prime number behaves randomly.
Yet, they behave regularly in aggregate.
4 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Moral of this talk
Each prime number behaves randomly.Yet, they behave regularly in aggregate.
4 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Randomness vs Regularity, according to Don Zagier
D. Zagier (1951–): There are two factsabout the distribution of prime numberswhich I hope to convince you sooverwhelmingly that they will bepermanently engraved in your hearts.
(continued on the next slide..)
5 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Zagier’s quote: continued
The first is that . . . the prime numbers belong to the most arbitraryand ornery objects studied by mathematicians: they grow like weedsamong the natural numbers, seeming to obey no other law than that ofchance, and nobody can predict where the next one will sprout.
The second fact is even more astonishing, for it states just theopposite: that the prime numbers exhibit stunning regularity, thatthere are laws governing their behavior, and that they obey these lawswith almost military precision.
In this talk, we focus on regularity, which becomes especiallyprominent when we try to count them.
6 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Zagier’s quote: continued
The first is that . . . the prime numbers belong to the most arbitraryand ornery objects studied by mathematicians: they grow like weedsamong the natural numbers, seeming to obey no other law than that ofchance, and nobody can predict where the next one will sprout.The second fact is even more astonishing, for it states just theopposite: that the prime numbers exhibit stunning regularity, thatthere are laws governing their behavior, and that they obey these lawswith almost military precision.
In this talk, we focus on regularity, which becomes especiallyprominent when we try to count them.
6 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Zagier’s quote: continued
The first is that . . . the prime numbers belong to the most arbitraryand ornery objects studied by mathematicians: they grow like weedsamong the natural numbers, seeming to obey no other law than that ofchance, and nobody can predict where the next one will sprout.The second fact is even more astonishing, for it states just theopposite: that the prime numbers exhibit stunning regularity, thatthere are laws governing their behavior, and that they obey these lawswith almost military precision.
In this talk, we focus on regularity, which becomes especiallyprominent when we try to count them.
6 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
There are enough supply of primes
Theorem (Euclid)
There are infinitely many primes.
Proof.Start with any finite list of primes {p1, p2, . . . , pn}. Then, thenumber
p1 · p2 · · · pn + 1
is not divisible by any prime in the list, therefore, contains anew prime number as its factor. So, it is always possible to addone more to the list! By iterating this process, you can create alist of primes as long as you wish. �
7 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
There are enough supply of primes
Theorem (Euclid)
There are infinitely many primes.
Proof.Start with any finite list of primes {p1, p2, . . . , pn}. Then, thenumber
p1 · p2 · · · pn + 1
is not divisible by any prime in the list, therefore, contains anew prime number as its factor. So, it is always possible to addone more to the list! By iterating this process, you can create alist of primes as long as you wish. �
7 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Prime counting function
Define π(x) by
π(x) = the number of primes p ≤ x.
We just proved that
π(x)→∞, as x→∞.
Question: How fast does π(x) go to infinity?
8 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The function π(x)
Question: How fast does π(x) go to infinity?
9 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The function π(x)
Question: How fast does π(x) go to infinity?
9 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The function π(x)
Question: How fast does π(x) go to infinity?
Computing π(x) exactly for large x is a very hard computationalchallenge. Current record of π(x) for the largest x is
π(1024) = 18435599767349200867866,
[Found by Buethe, Franke, Jost, Kleinjung in July 2010]
Asymptotic expression for π(x)
It would be nice to have a function, say f (x), with a closed-formformula such that
π(x) ∼ f (x)
(meaning: limx→1
π(x)/f (x) = 1.)
9 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The function π(x)
Question: How fast does π(x) go to infinity?
Computing π(x) exactly for large x is a very hard computationalchallenge. Current record of π(x) for the largest x is
π(1024) = 18435599767349200867866,
[Found by Buethe, Franke, Jost, Kleinjung in July 2010]
Asymptotic expression for π(x)
It would be nice to have a function, say f (x), with a closed-formformula such that
π(x) ∼ f (x) (meaning: limx→1
π(x)/f (x) = 1.)
9 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Guessing asymptotic formula with some data
x π(x) x/π(x)
10 4 2.5100 25 4.0
1,000 168 6.010,000 1,229 8.1100,000 9,592 10.4
1,000,000 78,498 12.710,000,000 664,579 15.0100,000,000 5,761,455 17.4
1,000,000,000 50,847,534 19.710,000,000,000 455,052,511 22.0100,000,000,000 4,118,054,813 24.3
10 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Guessing asymptotic formula with some data
x π(x) x/π(x)
10 4 2.5100 25 4.0
1,000 168 6.010,000 1,229 8.1100,000 9,592 10.4
1,000,000 78,498 12.710,000,000 664,579 15.0100,000,000 5,761,455 17.4
1,000,000,000 50,847,534 19.710,000,000,000 455,052,511 22.0100,000,000,000 4,118,054,813 24.3
x/π(x) ≈ (the # of digits in x) · 2.3
10 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Guessing asymptotic formula with some data
x π(x) x/π(x)
10 4 2.5100 25 4.0
1,000 168 6.010,000 1,229 8.1100,000 9,592 10.4
1,000,000 78,498 12.710,000,000 664,579 15.0100,000,000 5,761,455 17.4
1,000,000,000 50,847,534 19.710,000,000,000 455,052,511 22.0100,000,000,000 4,118,054,813 24.3
x/π(x) ≈ log10(x) · ln(10)
10 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Guessing asymptotic formula with some data
x π(x) x/π(x)
10 4 2.5100 25 4.0
1,000 168 6.010,000 1,229 8.1100,000 9,592 10.4
1,000,000 78,498 12.710,000,000 664,579 15.0100,000,000 5,761,455 17.4
1,000,000,000 50,847,534 19.710,000,000,000 455,052,511 22.0100,000,000,000 4,118,054,813 24.3
π(x) ∼x
ln(x)10 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Carl Friedrich Gauss
C. F. Gauss (1777–1855), when he was14 years old, conjectured that
π(x) ∼∫ x
2
dtln t
.
After he read a book that containedlogarithms of numbers up to 7 digitsand a table of primes up to 10,009.From the table of primes, he observedthat “the density (of prime numbers nearx) is, on the average, inverselyproportional to the logarithm of x”.
11 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Gauss’s guess on π(x)
Challenge for Calculus students
Prove thatlimx→∞
x/ ln x∫ x2 (1/ ln t) dt
= 1.
Hint: L’Hopital’s rule and Fundamental Theorem of Calculus!
Because of the above, Gauss’s formula
π(x) ∼∫ x
2
dtln t
is compatible with our formula
π(x) ∼x
ln x.
12 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Gauss’s guess on π(x)
Challenge for Calculus students
Prove thatlimx→∞
x/ ln x∫ x2 (1/ ln t) dt
= 1.
Hint: L’Hopital’s rule and Fundamental Theorem of Calculus!
Because of the above, Gauss’s formula
π(x) ∼∫ x
2
dtln t
is compatible with our formula
π(x) ∼x
ln x.
12 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Prime number theorem
Almost 100 years after Gauss stated his conjecture on π(x), theso-called Prime number theorem was finally proved:
Theorem (Hadamard and de la Vallee Poussin in 1896)As x→∞,
π(x) ∼x
ln x.
We will sketch some ideas behind the proof of this theorem.
13 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Proof of prime number theorem
Theorem (Prime number theorem)As x→∞,
π(x) ∼x
ln x.
Sketch of proof (adapted from presentation of T. Tao):
Artistic
1 Create a noise, ateach prime p.
2 Listen to the primesymphony.
3 (hardest) Writedown the notesyou’re listening.
Technical
1 Defineψ(x) :=
∑p≤x ln p.
2 Take (sort of) Fouriertransform of ψ(x).
3 (hardest) Locate thefrequencies.
14 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Proof of prime number theorem
Theorem (Prime number theorem)As x→∞,
π(x) ∼x
ln x.
Sketch of proof (adapted from presentation of T. Tao):
Artistic1 Create a noise, at
each prime p.
2 Listen to the primesymphony.
3 (hardest) Writedown the notesyou’re listening.
Technical
1 Defineψ(x) :=
∑p≤x ln p.
2 Take (sort of) Fouriertransform of ψ(x).
3 (hardest) Locate thefrequencies.
14 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Proof of prime number theorem
Theorem (Prime number theorem)As x→∞,
π(x) ∼x
ln x.
Sketch of proof (adapted from presentation of T. Tao):
Artistic1 Create a noise, at
each prime p.
2 Listen to the primesymphony.
3 (hardest) Writedown the notesyou’re listening.
Technical1 Defineψ(x) :=
∑p≤x ln p.
2 Take (sort of) Fouriertransform of ψ(x).
3 (hardest) Locate thefrequencies.
14 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Proof of prime number theorem
Theorem (Prime number theorem)As x→∞,
π(x) ∼x
ln x.
Sketch of proof (adapted from presentation of T. Tao):
Artistic1 Create a noise, at
each prime p.2 Listen to the prime
symphony.
3 (hardest) Writedown the notesyou’re listening.
Technical1 Defineψ(x) :=
∑p≤x ln p.
2 Take (sort of) Fouriertransform of ψ(x).
3 (hardest) Locate thefrequencies.
14 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Proof of prime number theorem
Theorem (Prime number theorem)As x→∞,
π(x) ∼x
ln x.
Sketch of proof (adapted from presentation of T. Tao):
Artistic1 Create a noise, at
each prime p.2 Listen to the prime
symphony.
3 (hardest) Writedown the notesyou’re listening.
Technical1 Defineψ(x) :=
∑p≤x ln p.
2 Take (sort of) Fouriertransform of ψ(x).
3 (hardest) Locate thefrequencies.
14 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Proof of prime number theorem
Theorem (Prime number theorem)As x→∞,
π(x) ∼x
ln x.
Sketch of proof (adapted from presentation of T. Tao):
Artistic1 Create a noise, at
each prime p.2 Listen to the prime
symphony.3 (hardest) Write
down the notesyou’re listening.
Technical1 Defineψ(x) :=
∑p≤x ln p.
2 Take (sort of) Fouriertransform of ψ(x).
3 (hardest) Locate thefrequencies.
14 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Proof of prime number theorem
Theorem (Prime number theorem)As x→∞,
π(x) ∼x
ln x.
Sketch of proof (adapted from presentation of T. Tao):
Artistic1 Create a noise, at
each prime p.2 Listen to the prime
symphony.3 (hardest) Write
down the notesyou’re listening.
Technical1 Defineψ(x) :=
∑p≤x ln p.
2 Take (sort of) Fouriertransform of ψ(x).
3 (hardest) Locate thefrequencies.
14 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
What is Fourier transform?
A (sample) soundwave:
(a periodic function)
(2) sin(x)
(−1) sin(2x)
(2/3) sin(3x)
2 sin(x) − sin(2x) +(2/3) sin(3x)
Fourier transform isa process of
decomposing aperiodic function
into several simplewaves of different
frequencies.
15 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
What is Fourier transform?
A (sample) soundwave:
(a periodic function)
(2) sin(x)
(−1) sin(2x)
(2/3) sin(3x)
2 sin(x) − sin(2x) +(2/3) sin(3x)
Fourier transform isa process of
decomposing aperiodic function
into several simplewaves of different
frequencies.
15 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
What is Fourier transform?
A (sample) soundwave:
(a periodic function)
(2) sin(x)
(−1) sin(2x)
(2/3) sin(3x)
2 sin(x) − sin(2x) +(2/3) sin(3x)
Fourier transform isa process of
decomposing aperiodic function
into several simplewaves of different
frequencies.
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
What is Fourier transform?
A (sample) soundwave:
(a periodic function)
(2) sin(x)
(−1) sin(2x)
(2/3) sin(3x)
2 sin(x) − sin(2x) +(2/3) sin(3x)
Fourier transform isa process of
decomposing aperiodic function
into several simplewaves of different
frequencies.
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Your ears do Fourier transform all the time.
When you hearsound like this..
Hairs in your earresonate...
Amplitude: 2
))
Amplitude: −1)Amplitude: 2/3
)
And, your braincollects the signals
from hairs!
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Your ears do Fourier transform all the time.
When you hearsound like this..
Hairs in your earresonate...
Amplitude: 2
))
Amplitude: −1)Amplitude: 2/3
)
And, your braincollects the signals
from hairs!
16 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Your ears do Fourier transform all the time.
When you hearsound like this..
Hairs in your earresonate...
Amplitude: 2
))
Amplitude: −1)Amplitude: 2/3
)
And, your braincollects the signals
from hairs!
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Visualizing Fourier transformation
Java applet demonstration:
http://www.falstad.com/fourier/
written by Paul Falstad.
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Georg Friedrich Bernhard Riemann
G. F. B. Riemann (1826–1866) madefundamental contributions in manybranches of mathematics. Especially,he was a pioneer of the theory offunctions, which is nowadays coveredin an undergraduate course oncomplex analysis.
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The Riemann’s zeta function ζ(s)
In 1864, Riemann wrote a short memoir, titled Uber die Anzahlder Primzahlen unter einer gegebenen Grosse.
(Translation: On theNumber of Primes Less than a Given Magnitude)
Riemann’s revolutionary idea
Study prime numbers by the zeta function
ζ(s) := 1 +12s +
13s +
14s + · · · ,
where s is a complex variable. (Simply put, a complex numbers = σ + it is just a pair (σ, t) of real numbers.)
Complex analysis techniques can show that the domain of ζ(s)can be extended to the entire complex plane, except s = 1.
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The Riemann’s zeta function ζ(s)
In 1864, Riemann wrote a short memoir, titled Uber die Anzahlder Primzahlen unter einer gegebenen Grosse. (Translation: On theNumber of Primes Less than a Given Magnitude)
Riemann’s revolutionary idea
Study prime numbers by the zeta function
ζ(s) := 1 +12s +
13s +
14s + · · · ,
where s is a complex variable. (Simply put, a complex numbers = σ + it is just a pair (σ, t) of real numbers.)
Complex analysis techniques can show that the domain of ζ(s)can be extended to the entire complex plane, except s = 1.
19 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The Riemann’s zeta function ζ(s)
In 1864, Riemann wrote a short memoir, titled Uber die Anzahlder Primzahlen unter einer gegebenen Grosse. (Translation: On theNumber of Primes Less than a Given Magnitude)
Riemann’s revolutionary idea
Study prime numbers by the zeta function
ζ(s) := 1 +12s +
13s +
14s + · · · ,
where s is a complex variable. (Simply put, a complex numbers = σ + it is just a pair (σ, t) of real numbers.)
Complex analysis techniques can show that the domain of ζ(s)can be extended to the entire complex plane, except s = 1.
19 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The Riemann’s zeta function ζ(s)
In 1864, Riemann wrote a short memoir, titled Uber die Anzahlder Primzahlen unter einer gegebenen Grosse. (Translation: On theNumber of Primes Less than a Given Magnitude)
Riemann’s revolutionary idea
Study prime numbers by the zeta function
ζ(s) := 1 +12s +
13s +
14s + · · · ,
where s is a complex variable. (Simply put, a complex numbers = σ + it is just a pair (σ, t) of real numbers.)
Complex analysis techniques can show that the domain of ζ(s)can be extended to the entire complex plane, except s = 1.
19 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The zeros σ + it of ζ(s)
σ
t
11/2
10
20
30
40
50 If ζ(σ + it) = 0, then we say that σ + it is azero of ζ(s). Riemann discovered:
ζ(s) has infinitely many zeros.
All (at least the interesting ones) areon the strip 0 ≤ σ ≤ 1.Riemann himself found a few zerosexplicitly (pictured left), all of whichhappen to be on the line σ = 1/2.Finally, Riemann conjectured that allthe zeros are on the line σ = 1/2.
The last item deserves some attention.
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The zeros σ + it of ζ(s)
σ
t
11/2
10
20
30
40
50 If ζ(σ + it) = 0, then we say that σ + it is azero of ζ(s). Riemann discovered:
ζ(s) has infinitely many zeros.All (at least the interesting ones) areon the strip 0 ≤ σ ≤ 1.
Riemann himself found a few zerosexplicitly (pictured left), all of whichhappen to be on the line σ = 1/2.Finally, Riemann conjectured that allthe zeros are on the line σ = 1/2.
The last item deserves some attention.
20 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The zeros σ + it of ζ(s)
σ
t
11/2
10
20
30
40
50 If ζ(σ + it) = 0, then we say that σ + it is azero of ζ(s). Riemann discovered:
ζ(s) has infinitely many zeros.All (at least the interesting ones) areon the strip 0 ≤ σ ≤ 1.Riemann himself found a few zerosexplicitly (pictured left), all of whichhappen to be on the line σ = 1/2.
Finally, Riemann conjectured that allthe zeros are on the line σ = 1/2.
The last item deserves some attention.
20 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The zeros σ + it of ζ(s)
σ
t
11/2
10
20
30
40
50 If ζ(σ + it) = 0, then we say that σ + it is azero of ζ(s). Riemann discovered:
ζ(s) has infinitely many zeros.All (at least the interesting ones) areon the strip 0 ≤ σ ≤ 1.Riemann himself found a few zerosexplicitly (pictured left), all of whichhappen to be on the line σ = 1/2.Finally, Riemann conjectured that allthe zeros are on the line σ = 1/2.
The last item deserves some attention.
20 / 29
Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
The zeros σ + it of ζ(s)
σ
t
11/2
10
20
30
40
50 If ζ(σ + it) = 0, then we say that σ + it is azero of ζ(s). Riemann discovered:
ζ(s) has infinitely many zeros.All (at least the interesting ones) areon the strip 0 ≤ σ ≤ 1.Riemann himself found a few zerosexplicitly (pictured left), all of whichhappen to be on the line σ = 1/2.Finally, Riemann conjectured that allthe zeros are on the line σ = 1/2.
The last item deserves some attention.
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Riemann Hypothesis
The last item (the conjecture thatall the zeta zeros are on σ = 1/2) iscalled Riemann Hypothesis. Itis now literally a million-dollarquestion, thanks to the bountyoffered by the ClayMathematics Institute.
So, what do the zeta zeroes have to do with prime-countingand Fourier analysis?
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Riemann Hypothesis
The last item (the conjecture thatall the zeta zeros are on σ = 1/2) iscalled Riemann Hypothesis. Itis now literally a million-dollarquestion, thanks to the bountyoffered by the ClayMathematics Institute.
So, what do the zeta zeroes have to do with prime-countingand Fourier analysis?
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
What Riemann heard
σ
t
11/2
HHJL
Each zero of ζ(s) corresponds to oneparticular frequency, or one note.Riemann had good enough ears torecord a few notes in the primesymphony.To finish the proof of Prime numbertheorem, one needs to prove thatthere is no note on the line σ = 1,which was technically the hardestpart, and took almost 30 years afterRiemann.
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Prime symphony
Since Riemann, number theorists have understood that, themore we know about the locations of zeros of the Riemann zetafunction ζ(s), the better we understand the distribution ofprime numbers.
Artistically speaking, this means that, to best understand thedistribution of prime numbers, we need to write down as manynotes as possible, while listening to the prime symphony. (Recall:there are infinitely many notes.)
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Prime symphony
Since Riemann, number theorists have understood that, themore we know about the locations of zeros of the Riemann zetafunction ζ(s), the better we understand the distribution ofprime numbers.
Artistically speaking, this means that, to best understand thedistribution of prime numbers, we need to write down as manynotes as possible, while listening to the prime symphony. (Recall:there are infinitely many notes.)
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
A maestro, who has (computationally) perfect pitch
In the 80’s, A. Odlyzko, who is amaster in various number theoreticalcomputations, succeeded in locatingbillions of zeta zeros. His data gave avery strong statistical support to anearlier stunning observation made byMontgomery in early 70’s, which was. . .
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Prime symphony sounds a lot like... heavy nuclei?
Montgomery-Odlyzko Law
The distribution of zeta zeros statistically follows that of theenergy levels of heavy nuclei.
In order to describe the energy levels of heavy nuclei, whoseexact determination is virtually impossible, physicists use astatistical model, called GUE (Gaussian Unitary Ensemble),derived from the distribution of eigenvalues of large unitarymatrices.
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Zeta zeros vs GUE, I
100 zeta zeros starting att = 1200, wrapped around a
unit circle once
100 eigenvalues of a random100 × 100 unitary matrix (the
GUE model)
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Zeta zeros vs GUE, II
Nearest neighborspacings among1000 zeta zeroes,versus the GUE
model
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Zeta zeros vs GUE, II
Nearest neighborspacings among 70
million zetazeroes, versus the
GUE model
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Zeta zeros vs GUE, II
Nearest neighborspacings among 70
million zetazeroes, versus the
GUE model
P. Diaconis (statistician): “I’ve been a statistician all my life, andI’ve never seen such a good fit of data.”
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Zeta zeros vs GUE, II
Nearest neighborspacings among 70
million zetazeroes, versus the
GUE model
Katz and Sarnak (number theorists): “At the phenomenologicallevel this is perhaps the most striking discovery about zeta sinceRiemann.”
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Introduction Prime number theorem Proof of prime number theorem Riemann’s idea Prime symphony
Number theory and physics
This unexpected connection between number theory andphysics has recently generated a lot of interesting research andcontinued to be a prime topic in modern number theory.
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