## Distribution of primes and a weighted energy problem

Igor E. Pritsker

### Abstract

We discuss a recent development connecting the asymptotic distribution of prime numbers with weighted potential theory. These ideas originated with the Gelfond-Schnirelman method (circa 1936), which used polynomials with integer coefficients and small sup norms on $[0,1]$ to give a Chebyshev-type lower bound in prime number theory. A generalization of this method for polynomials in many variables was later studied by Nair and Chudnovsky, who produced tight bounds for the distribution of primes. Our main result is a lower bound for the integral of Chebyshev's $\psi$-function, expressed in terms of the weighted capacity for polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. This new connection leads to some interesting open problems on weighted capacity.

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### Key words

distribution of prime numbers, polynomials, integer coefficients, weighted transfinite diameter, weighted capacity, potentials

### AMS subject classifications

11N05, 31A15, 11C08