The Prime Number Theorem Yusuf Chebao The main purpose of these notes is to present a fairly readable version of a proof of the Prime Number Theorem (PNT), expanded from Sections 17-18 of Davenport's text [3]. We intend to make this exposition as self-contained as possible. However, some results from earlier sections in the text, as well as some principles from complex analysis, are assumed and listed without proofs in Section 1 below. In Section 2, we derive a formula for the sum of the von Mangoldt function, which is a key ingredient of our proof. The final steps of the proof are carried out in Section 3, where some connection between the PNT and the Riemann Hypothesis (RH) is also discussed. Let us first recall a few definitions: • The prime counting function π(x) is the number of all primes less than or equal to a positive real number x. • The logarithmic integral function is Z x 1 li(x) = dt: 2 log t Note that some authors denote this function as Li(x). Here we follow the convention by Davenport. • The big O notation: Let f; g and h be complex-valued functions whose domains contain a common infinite interval [a; 1). ◦ The relation f(x) = O(g(x)) means there exist some constants M and x0 such that jf(x)j ≤ Mjg(x)j for all x ≥ x0. ◦ The relation f(x) = h(x) + O(g(x)) means f(x) − h(x) = O(g(x)). This is to be understood that f(x) can be approximated by h(x), where g(x) is an error term. • We occasionally write f(x) ≪ g(x) for f(x) = O(g(x)). The form of the PNT that we will be proving is ( ) x π(x) = li(x) + O p (1) ec log x where c is a positive constant. We will also show that an immediate consequence of (1) is another form of the PNT: ( ) x x π(x) = + O : (2) log x log2 x 2 Several proofs of the PNT have been discovered. Significant contributions that lead to the first successful proof could be traced back at least to the 18th century, since the time of Legendre and Gauss. Below we record a number of notorious achievements and some well-known proofs of the PNT, listed by year. We do not intend to produce an exhaustive timeline nor a complete historical account here. 1851 Pafnuty Chebyshev showed that there exist constants c1 ≈ 0:9 and ≈ x ≤ ≤ x c2 1:1 such that c1 log x π(x) c2 log x for all large x. The preceding ≍ x inequalities are abbreviated as π(x) log x , and can be read as \π(x) x has the same order of magnitude as log x ". Thus Chebyshev was the first to demonstrate the correct order of magnitude for π(x). See Section 3.1 of [6] for a detailed discussion of Chebyshev's estimates. 1859 Bernhard Riemann, in his influential memoir \On the number of primes less than a given magnitude", used the zeta function ζ(s) which was extended analytically throughout the complex plane (except for a pole at s = 1), to investigate π(x). Among other things, he established an important functional equation for ζ(s), namely, ξ(s) = ξ(1 − s), − s s 2 where ξ(s) = π Γ( 2 )ζ(s) and Γ is the gamma function. In addition, he showed some connection between π(x) and the zeros of ζ(s). This paved a way to the first complete proof of the PNT a few decades later. 1896 Jacques Hadamard and Charles Jean de la Vall´ee-Poussin in- dependently gave the first complete proof of the PNT. The proof uses techniques from complex analysis, particularly the Argument Principle and the Residue Theorem, and is based on some careful analysis of the zeros of ζ(s), as suggested earlier by Riemann. This proof is sometimes referred to as an analytic proof of the PNT. 1931 Shikao Ikehara, in [7], used Wiener's Tauberian Theorem to derived the PNT. An outstanding aspect of this proof is that it only requires ζ(s) to be zero-free on the line Re(s) = 1, rather than a larger zero-free region as in the previous proof by Hadamard and de la Vall´ee-Poussin. 1949 Paul Erd}os in [4] and Atle Selberg in [9] published essentially the same elementary proof of the PNT. It is elementary in the sense that no sophisticated technique from complex analysis was used, although the argument was quite intricate. Unfortunately, there was a bitter disagreement between the two mathematicians as to whom the credit of the proof should be given. See [5] for an account of the dispute. 1980 Donald J. Newman found a simple proof of the PNT. It is analytic, but simple in a sense that it hardly uses anything beyondP Cauchy's Integral Formula, and uses classical results on θ(x) = p≤x log p by Chebyshev. It can also be viewed as a simpler version of the Tauberian argument by Ikehara. This is perceived by some as the easiest proof the PNT so far. See [1] or [8] for Newman's proof. Our proof of the PNT will follow essentially a classical approach by Hadamard and de la Vall´ee-Poussin. 3 The following diagram illustrates logical dependence of the main results in these notes. An arrow from A to B indicates that we use A (possibly along with some other results) to prove B. Theorems 1-2 Theorem 3 Theorem 4 Theorems 5-6 / Remark 7 XXXXX QQQ { XXXXX QQQ {{ XXXXX QQ { XXXXX QQQ {{ XXXXX QQ( {{ , {{ / / / {{ Theorem 8 Theorem 10 TheoremQQ 11Q Corollary 12 { QQQ {{ QQQ {{ QQQ {{ QQ( {} { Corollary 9 Theorem 13 / Theorem 14 Theorem 1 and Theorem 2 are basic principles about contour integrals that we will use to derive two formulas for the sum of von Mangoldt function in Theorem 10 and Theorem 11. Notice that Theorem 11 is a central result that is also based on Theorem 3 and Theorem 4, which are some facts about the gamma function and the zeta function. Theorem 13 is a crucial estimate for the sum of von Mangoldt function that will eventually yield Theorem 14, which is the PNT. 1 Preliminaries In this section, we collect some results (mostly without proofs) that are required for the next sections. First of all, in the derivations of the error terms, we almost always rely on the following fact about upper bounds for the moduli of contour integrals: Theorem 1. Let C := z(t); t 2 [a; b] be a smooth arc and let f(z) be a continuous function on C. Then Z Z b f(x)dz ≤ jf(z(t))jjdz(t)j ≤ ML C R a where M := max jf(z(t))j and L = b jz0(t)jdt is the length of C. t2[a;b] a R j j ≤ The relation C f(z)dz ML is called the ML inequality. Another important principle that we will be using repeatedly in Section 2 is: Theorem 2. (The Residue Theorem) Let C be a simple closed curve in a region A, traversed once in the counterclock- wise direction. Let f be a complex-valued function which is analytic on and inside C, except at some isolated singularities z1; : : : ; zk inside C. Then Z 1 Xk f(z)dz = Res(f; z ): 2πi j C j=i For the proofs of the two theorems above, see, for example, [1] or [2]. Further, we will extensively use many facts about the Riemann zeta function ζ(s), which is closely related to the gamma function Γ(s). 4 • The gamma function can be defined on C − f0; −1; −2;::: g by 1 [( ) ] e−γs Y s −1 Γ(s) = 1 + es=n s n n=1 ( P ) where γ = lim n 1 − log n is the Euler constant. It is known that Γ(s) is n!1 k=1 k zero-free and analytic in the complex plane, except at simple poles s = 0; −1; −2; −3;::: . In Section 2, we will need a fact about the logarithmic derivative of the gamma function that Γ0(s) = log s + O(jsj−1) (3) Γ(s) when jsj ! 1 and −π + δ < arg(s) ≤ π − δ for any δ > 0. This fact is taken from Section 10 of [3]. P • 1 s The Riemann zeta function is defined for Re(s) > 1 by ζ(s) = n=1 1=n , and can be extended to the right half-plane Re(s) > 0 by Z s 1 fxg ζ(s) = − s dx − s+1 s 1 1 x where fxg = x − bxc is the fractional part of x. One way to extend ζ(s) to the whole complex plane is to use the following form of Riemann's functional equation: ( ) πs ζ(s) = 2sπs−1ζ(1 − s)Γ(1 − s) sin : (4) 2 We know that ζ(s) is analytic in C, except for a simple pole at s = 1. It can be seen from (4) that the only zeros of ζ(s) on the left half-plane are s = −2; −4; −6;::: ; these are called the trivial zeros of the Riemann zeta function, and each of them has order one. All the remaining zeros of ζ(s) are in the critical strip fs 2 C : 0 < Re(s) < 1g, and are called nontrivial zeros. The Riemann Hypothesis (RH) asserts that 1 all nontrivial zeros of ζ(s) are on the critical line Re(s) = .