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The Gelfond–Schnirelman method in prime number theory
 Canad. J. Math
, 2005
"... Abstract. The original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lowe ..."
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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 Chebyshevtype 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 ψfunction, 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 polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1 Lower Bounds for Arithmetic Functions Let π(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that (1.1) π(x) ∼ x log x as x → ∞. We include a very brief sketch of its history, referring for details to many excellent books and surveys available on this subject (see, e.g., [8, 10, 17, 29]). Chebyshev [6] made the first important step towards the PNT in 1852, by proving the bounds (1.2) 0.921 x log x
DISTRIBUTION OF PRIMES AND A WEIGHTED ENERGY PROBLEM
"... Abstract. We discuss a recent development connecting the asymptotic distribution of prime numbers with weighted potential theory. These ideas originated with the GelfondSchnirelman method (circa 1936), which used polynomials with integer coefficients and small sup norms on ¢ £¥¤§¦© ¨ to give a Cheb ..."
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Abstract. We discuss a recent development connecting the asymptotic distribution of prime numbers with weighted potential theory. These ideas originated with the GelfondSchnirelman method (circa 1936), which used polynomials with integer coefficients and small sup norms on ¢ £¥¤§¦© ¨ to give a Chebyshevtype lower bound in prime number theory. A generalization of this method for polynomials in many variables was later studied by Nair and Chudnovsky, who produced tight bounds for the distribution of primes. Our main result is a lower bound for the integral of Chebyshev’s �function, expressed in terms of the weighted capacity for polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. This new connection leads to some interesting open problems on weighted capacity.