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The Gelfond–Schnirelman Method in Prime Number Theory

Published online by Cambridge University Press:  20 November 2018

Igor E. Pritsker
Igor E. Pritsker*
Affiliation:
Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, OK 74078-1058, U.S.A., e-mail: igor@math.okstate.edu
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Abstract

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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

Primary: 11N0531A15secondary: 11C08distribution of prime numberspolynomialsinteger coefficientsweighted transfinite diameterweighted capacitypotentials
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 5 , 01 October 2005 , pp. 1080 - 1101
Copyright
Copyright © Canadian Mathematical Society 2005

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