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The Gelfond--Schnirelman Method in Prime Number Theory |
Canad. J. Math. 57(2005), 1080-1101
Published:2005-10-01
Printed: Oct 2005
Printed: Oct 2005
- Igor E. Pritsker
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] |