The Gelfond-Schnirelman method in prime number theory

TL;DR

This paper generalizes the Gelfond-Schnirelman method to multivariable polynomials, establishing new lower bounds for prime number distribution using potential theory and extremal measures.

Contribution

It extends the classical method to multiple variables and solves the associated potential theoretic problem, linking prime bounds to weighted capacity and extremal measures.

Findings

Derived a lower bound for Chebyshev's ψ-function using weighted capacity.

Solved the extremal measure and support in the potential theoretic problem.

Connected polynomial weights with prime number bounds via potential theory.

Abstract

The original Gelfond-Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on $[0,1]$ to give a Chebyshev-type 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 $\psi$-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 polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support.