Simple Proof of the Prime Number Theorem , etc

The point here is the relatively simple argument that non-vanishing of an L-function on the line Re (s) = 1 implies an asymptotic result parallel to the application of ζ(s) to the Prime Number Theorem. This is based upon [Newman 1980]. In particular, this argument avoids estimates on the zeta function at infinity and also avoids Tauberian arguments. For completeness, we recall the standard clever ad hoc argument for the non-vanishing of ζ(s) on Re (s) = 1, thus giving a complete proof of the Prime Number Theorem here. However, the larger intent is to prove non-vanishing results for L-functions by capturing the L-functions in constant terms of Eisenstein series (after Langlands and Shahidi), and then apply the present argument to obtain the most immediate asymptotic corollary. • Non-vanishing of L-functions on Re(s) = 1 • Convergence theorem • First corollary on asymptotics • Elementary lemma on asymptotics • The Prime Number Theorem • Second corollary on asymptotics • A general asymptotic result 1. Non-vanishing of L-functions on Re(s) = 1 As the simplest example, the Riemann zeta function ζ(s) = n 1 n s = p prime 1 1 − 1 p s does not vanish on the line Re(s) = 1. This is not obvious! (The usual simple but ad hoc proof is given just below, for completeness.) As a consequence, using the Euler product expansion over primes, its logarithmic derivative d ds log ζ(s) = ζ (s) ζ(s) = − p d ds log(1 − p −s) = − p log p p s − p m≥2 log p p ms is holomorphic (except for the pole at s = 1) on an open set containing Re (s) ≥ 1. From this we prove (below) the Prime Number Theorem lim x→∞ number of primes ≤ x x/ log x = 1 or, at it is usually written, π(x) ∼ x log x As is well known, a form of this was conjectured by Gauss, and the theorem was proven independently by Hadamard and by de la Valleé Poussin. The methodology below is perhaps the clearest proof of the Prime Number Theorem, using simplifications found by D.J. Newman about 1980. However, the simplified form does not give any indication of the relation between zero-free regions and the error term in the Prime Number Theorem. In general, non-vanishing of an L-function (with Euler product) on a vertical line implies …