The original Gelfond-Schnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.

The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number x>1, let π(x) denote the number of primes less than x. The prime number theorem is the assertion that lim x→ ∞ π(x) x

In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zeta-function.

An exposition is given, partly historical and partly mathematical, of the Riemann zeta function ζ(s) and the associated Riemann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count non-trivial zeros of ζ(s). Relevance of these investigations to the theory of the distribution of prime numbers is discussed.

The key result needed in almost all elementary proofs of the Prime Number Theorem is a prime number lemma proved by Atle Selberg in 1948. Without restricting ourselves to purely elementary techniques we show how the error term in Selberg’s fundamental lemma relates to the error term in the Prime Number Theorem. In spite of all the interest in this topic over the last sixty years this particular question seems to have been overlooked in the past.

Riemann hypothesis, universality & broken symmetry