The Prime Number Theorem A PRIMES Exposition Ishita Goluguri, Toyesh Jayaswal, Andrew Lee Mentor: Chengyang Shao TABLE OF CONTENTS 1 Introduction 2 Tools from Complex Analysis 3 Entire Functions 4 Hadamard Factorization Theorem 5 Riemann Zeta Function 6 Chebyshev Functions 7 Perron Formula 8 Prime Number Theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 2 Introduction • Euclid (300 BC): There are infinitely many primes • Legendre (1808): for primes less than 1,000,000: x π(x) ' log x © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 3 Progress on the Distribution of Prime Numbers • Euler: The product formula 1 X 1 Y 1 ζ(s) := = ns 1 − p−s n=1 p so (heuristically) Y 1 = log 1 1 − p−1 p • Chebyshev (1848-1850): if the ratio of π(x) and x= log x has a limit, it must be 1 • Riemann (1859): On the Number of Primes Less Than a Given Magnitude, related π(x) to the zeros of ζ(s) using complex analysis • Hadamard, de la Vallée Poussin (1896): Proved independently the prime number theorem by showing ζ(s) has no zeros of the form 1 + it, hence the celebrated prime number theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 4 Tools from Complex Analysis Theorem (Maximum Principle) Let Ω be a domain, and let f be holomorphic on Ω. (A) jf(z)j cannot attain its maximum inside Ω unless f is constant. (B) The real part of f cannot attain its maximum inside Ω unless f is a constant. Theorem (Jensen’s Inequality) Suppose f is holomorphic on the whole complex plane and f(0) = 1. Let Mf (R) = maxjzj=R jf(z)j. Let Nf (t) be the number of zeros of f with norm ≤ t where a zero of multiplicity n is counted n times. Then R Z Nf (t) dt ≤ log Mf (R): 0 t • Relates growth of a holomorphic function to distribution of its zeroes • Used to bound the number of zeroes of an entire function © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 5 Theorem (Borel-Carathéodory Lemma) Suppose f = u + iv is holomorphic on the whole complex plane. Suppose u ≤ A on @B(0;R): Then 2n! jf (n)(0)j ≤ (A − u(0)) Rn • Bounds all derivatives of f at 0 using only the real part of f • Used in proof of Hadamard Factorization Theorem to prove that function is a polynomial by taking limit and showing that nth derivative approaches 0 © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 6 Entire Functions Definition (Order) The order of an entire function, f, is the infimum of all possible λ > 0 such that there exists constants A and B that satisfy λ jf(z)j ≤ AeBjzj • sin z, cos z have order 1 p • cos z has order 1=2 © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 7 Entire Functions Theorem Let f be an entire function of order λ with f(0) = 1. Then, for any " > 0 there exists a constant, C", that satisfies λ+" Nf (R) ≤ C"R Theorem Let f be an entire function of order λ with f(0) = 1 and a1; a2; ::: be the zeroes of f in non-decreasing order of norms. Then, for any " > 0, 1 X 1 < 1 ja jλ+" n=1 n In other words, the convergence index of the zeros is at most λ. Forp example, sin z and cos z have order 1, and their zeroes grow linearly while cos z has order 1=2, and its zeroes grow quadratically. © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 8 Hadamard Factorization Theorem Theorem (Hadamard Factorization Theorem) A complex entire function f(z) of finite order λ and roots ai can be written as 1 p ! Y z X zk f(z) = eQ(z) 1 − exp a kak n=1 n k=1 n with p = bλc, and Q(z) being some polynomial of degree at most p The theorem extends the property of polynomials to be factored based on their roots as n Y z k 1 − : ai i=1 © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 9 proof The proof is based off of truncating the first p terms of the series 1 z X zk log 1 − = − k an ka k=1 n which bounds the magnitude to O(Rλ+) and gives rise to the exponential factor. Now, 1 p ! Y z X zk g(z) = 1 − exp k an ka n=1 k=1 n has the same roots as f(z) and the polynomial Q(z) is found by taking the logarithm of f(z)=g(z). The degree of Q is determined by bounding the derivatives using the Borel-Carathéodory lemma. © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 10 Riemann Zeta Function Definition (Reimann ζ Function) 1 X 1 ζ(s) = ; s = σ + it; σ > 1: ns n=1 Theorem (Euler Product Formula) The zeta function can also be written as Y 1 ζ(s) = −s p 1 − p The Euler product formula is the analytic equivalent of the unique factorization theorem for integers. © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 11 Chebyshev Functions Definition (Van-Mangoldt Function) ( log p if n = pm for some prime p and some m ≥ 1 Λ(n) = 0 otherwise Definition (Chebyshev Functions) X X (x) = Λ(n);#(x) = log p: n≤x p≤x Theorem 1 ζ0 X Λ(n) (s) = − ζ ns n=1 © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 12 Equivalent Asymptotic Expressions The Chebyshev functions can be related to π(x) by the following integral expressions. Theorem Z x π(t) #(x) = π(x) log x + dt 2 t #(x) Z x #(t) π(x) = + 2 dt log x 2 t log t Studying the asymptotic behavior of the formulas, we see that all of the following expressions are logically equivalent: π(x) log x lim = 1 x!1 x #(x) lim = 1 x!1 x (x) lim = 1 x!1 x © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 13 Perron Formula The Perron Formula acts as a filter to isolate the first finitely many terms from a Dirichlet series. Theorem (Perron Formula) P1 s Let F (s) = n=1 f(n)=n be absolutely convergent for σ > σa. Then for arbitrary c; x > 0, if σ > σa − c, 1 Z c+1i xz X f(n) F (s + z) dz = ∗ 2πi z ns c−∞i n≤x where P∗ means that the last term is halved when x is an integer. Corollary For x > 0, Z x (y) 1 Z 2+i1 ζ0 xs dy = − (s) ds 2 0 y 2πi 2−i1 ζ s © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 14 Reimann Zeta Function Continued We define a function ξ in terms of Γ; ζ and π to obtain a functional equation that gives information about the symmetry of zero distribution. Definition Γ s ζ(s) ξ(s) = 2 : πs=2 Theorem (Functional Equation, Riemann 1859) ξ(s) = ξ(1 − s) The proof relies on the Poisson summation formula from Fourier analysis. © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 15 Proof of the Functional Equation s 1 Z 1 Γ ζ(s) X s=2−1 −n2πx 2 = x e dx πs=2 n=1 0 Z 1 Z 1 Z 1 = xs=2−1Ψ(x)dx = + 0 0 1 where 1 X −πn2x Ψ(x) = e : n=1 Using the Poisson summating formula, 1 1 2Ψ(x) + 1 = p 2Ψ + 1 ; x x substituting into the integral from 0 to 1, 1 Z 1 1 Z 1 ξ(s) = + xs=2−3=2Ψ dx + xs=2−1Ψ(x)dx: s(s − 1) 0 x 1 © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 16 1 Changing the variable x ! x s Z 1 Γ 2 ζ(s) 1 −s=2−1=2 s=2−1 ξ(s) = s=2 = + x + x Ψ(x)dx π s(s − 1) 1 Substituting s = 1 − s gives ξ(s) = ξ(1 − s). © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 17 p-adic Analysis • The functional relations of ζ(s) can remarkably be obtained by studying the theory of p-adic numbers. • Generally, the distance between two numbers is considered using the usual metric jx − yj, but for every prime p, a separate notion of distance can be made for Q. n • For a rational number x = p a=b, p - a; b, we define the p-adic absolute value −n as jxjp = p . Then, the p-adic distance between two numbers is defined as jx − yjp. • The p-adic absolute value is multiplicative, positive definite, and satisfies the strong triangle inequality: jx − yjp ≤ max(jxjp; jyjp) ≤ jxjp + jyjp • Examples: j7=24j2 = 8 j2 − 27j5 = 1=25 © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 18 Qp • Q can be completed under this metric to form the field Qp • Qp contains a subring, Zp, which is the completion of Z under j · jp • Zp can be written using an "infinite p-adic expansion" with no negative powers and numbers in Qp have finite negative powers • Example: in Q5, 1 2 X n 1 1 ::: + 5 + 5 + 1 = 5 = = − 2 1 − 5 4 Z5 n=0 4 and −1=5 = :::111 × 5 = :::444:4 × × S n × • Letting p denote the elements of p for which jxjp = 1, p = p p Z Z Q n2Z Z © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 19 Adèles and Idèles • The fields R and Qp can be put together to form what is called the ring of Adèles • The Adèle ring AQ is the collection of sequences x = fxpgp2P[f1g for the primes P and where for each p 2 P, xp 2 Qp and x1 2 R with almost all xp 2 Zp × • The Idèle group IQ is the collection of Adèles for which almost all xp 2 Zp and forms a group under componentwise multiplication • We can also introduce a topology on IQ with open sets being the product of × × open sets in R and Qp , making IQ is a locally compact abelian group, a Haar measure, µ, can be formed for it Q • Introduce the volume function kxk := jx1j × jxpjp and the function p2P 2 Q '(x) := exp(−πjx1j ) 1Zp (xp) and consider the integral Z '(x)kxksdµ(x) IQ © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 20 The ξ Function Revisited The integral can be split into the real and p-adic components to obtain Z Z ! −πjtj2 s dt Y s × ξ(s) = e jtj × jxpjpdµp (xp) : × t × R p Zp s −s=2 1 The first integral turns out to be Γ 2 π and the latter ones are 1−p−s which combine to form ζ(s).