Newman's Short Proof of the Prime Number Theorem

Abstract

The prime number theorem, that the number of primes < x is asymptotic to x/log x, was proved (independently) by Hadamard and de la Vallee Poussin in 1896. Their proof had two elements: showing that Riemann's zeta function ;(s) has no zeros with Sc(s) = 1, and deducing the prime number theorem from this. An ingenious short proof of the first assertion was found soon afterwards by the same authors and by Mertens and is reproduced here, but the deduction of the prime number theorem continued to involve difficult analysis. A proof that was elementaty in a technical sense-it avoided the use of complex analysis-was found in 1949 by Selberg and Erdos, but this proof is very intricate and much less clearly motivated than the analytic one. A few years ago, however, D. J. Newman found a very simple version of the Tauberian argument needed for an analytic proof of the prime number theorem. We describe the resulting proof, which has a beautifully simple structure and uses hardly anything beyond Cauchy's theorem. Recall that the notation f(x) g(x) (f and g are asymptotically equal)