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The Gelfond–Schnirelman Method in Prime Number Theory
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- Journal:
- Canadian Journal of Mathematics / Volume 57 / Issue 5 / 01 October 2005
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1080-1101
- Print publication:
- 01 October 2005
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- Article
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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.
