The Gelfond–Schnirelman Method in Prime Number Theory

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 thismethod for polynomials inmany variables. Ourmain result is a lower bound for the integral of Chebyshev's ψ-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.

Zero distribution of sequences of classical orthogonal polynomials

We obtain the zero distribution of sequences of classical orthogonal polynomials associated with Jacobi, Laguerre, and Hermite weights. We show that the limit measure is the extremal measure associated with the corresponding weight.

Finite-infinite range inequalities in the complex plane

LetE⫅Cbe closed,ωbe a suitable weight function onE,σbe a positive Borel measure onE. We discuss the conditions onωandσwhich ensure the existence of a fixed compact subsetKofEwith the following property. For anyp,0<P≤∞, there exist positive constantsc1, c2depending only onE,ω,σandpsuch that for every integern≥1and every polynomialPof degree at mostn,∫E\K|ωnP|pdσ≤c1exp(−c2n)∫K|ωnP|pdσ. In particular, we shall show that the support of a certain extremal measure is, in some sense, the smallest setKwhich works. The conditions onσare formulated in terms of certain localized Christoffel functions related toσ.