Conditional mean values of long Dirichlet polynomials

TL;DR

Under the Riemann hypothesis, the paper derives asymptotic mean value formulas and large deviation estimates for long Dirichlet polynomials involving the von Mangoldt function, without relying on correlation sum estimates.

Contribution

It provides new asymptotic formulas and deviation estimates for long Dirichlet polynomials under specific weight conditions, avoiding correlation sum techniques.

Findings

Asymptotic mean value formulas for long Dirichlet polynomials established

Large deviation estimates derived for these polynomials

Results depend on specific weight conditions and the Riemann hypothesis

Abstract

Conditionally on the Riemann hypothesis we prove asymptotic formulae for mean values of various long Dirichlet polynomials involving the von Mangoldt function. Our results avoid the use of correlation sum estimates although in addition to the Riemann hypothesis we must assume that our Dirichlet polynomials have weights from a specific class whose transforms are sufficiently concentrated near the origin. We also give large deviation estimates for these long Dirichlet polynomials.