Cauchy Means of Dirichlet polynomials

TL;DR

This paper investigates Cauchy means of Dirichlet polynomials, establishing optimal bounds, extending results to new parameter ranges, and connecting classical methods with probabilistic approaches for improved analysis.

Contribution

The paper extends Wilf's analysis of Cauchy means of Dirichlet polynomials by providing new estimates, proving optimal bounds, and integrating probabilistic methods for broader applicability.

Findings

Established the optimality of certain upper bounds.

Derived new estimates for parameters q ≥ 1, σ ≥ 0, s > 0.

Connected integral operator theory with Brownian motion approaches.

Abstract

We study Cauchy means of Dirichlet polynomials $$\int_\R \Big|\sum_{n=1}^N \frac{1}{ n^{\s+ ist}} \Big|^{2q} \frac{\dd t}{\pi( t^2+1)}.$$ These integrals were investigated when $q=1,\s= 1, s=1/2 $ by Wilf, using integral operator theory and Widom's eigenvalue estimates. We show the optimality of some upper bounds obtained by Wilf. We also obtain new estimates for the case $q\ge 1$, $\s\ge 0$ and $s>0$. We complete Wilf's approach by relating it with other approaches (having notably connection with Brownian motion), allowing simple proofs, and also prove new results.