CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

The Gelfond--Schnirelman Method in Prime Number Theory

  Published:2005-10-01
 Printed: Oct 2005
  • Igor E. Pritsker
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

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 distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials
MSC Classifications: 11N05, 31A15, 11C08 show english descriptions Distribution of primes
Potentials and capacity, harmonic measure, extremal length [See also 30C85]
Polynomials [See also 13F20]
11N05 - Distribution of primes
31A15 - Potentials and capacity, harmonic measure, extremal length [See also 30C85]
11C08 - Polynomials [See also 13F20]
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/