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# The Gelfond--Schnirelman Method in Prime Number Theory

Published:2005-10-01
Printed: Oct 2005
• Igor E. Pritsker
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## 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.
 Keywords: distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials
 MSC Classifications: 11N05 - Distribution of primes 31A15 - Potentials and capacity, harmonic measure, extremal length [See also 30C85] 11C08 - Polynomials [See also 13F20]

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