An inequality of Hardy--Littlewood type for Dirichlet polynomials

TL;DR

This paper establishes a Hardy--Littlewood type inequality for Dirichlet polynomials, providing bounds on their $L^q$ norms and applying these to classical and conjectural Dirichlet series, including $L$-functions.

Contribution

It introduces a new inequality relating the coefficients of Dirichlet polynomials to their $L^q$ norms, extending classical results and applying to a broad class of Dirichlet series.

Findings

Derived an inequality for $L^q$ norms of Dirichlet polynomials for $0<q<2$.

Established asymptotic bounds for the $L^q$ norm of the specific Dirichlet series $D_N(s)$.

Extended bounds to a large class of Dirichlet series, including the Selberg class of $L$-functions.

Abstract

The $L^q$ norm of a Dirichlet polynomial $F(s)=\sum_{n=1}^{N} a_n n^{-s}$ is defined as \[\| F\|_q:=(\lim_{T\to\infty}\frac{1}{T}\int_{0}^T |F(it)|^qdt)^{1/q}\] for $0<q<\infty$. It is shown that \[ (\sum_{n=1}^{N} |a_n|^2|\mu(n)|[d(n)]^{\frac{\log q}{\log 2} -1})^{1/2}\le \| F\|_q \] when $0<q<2$; here $\mu$ is the M\"{o}bius function and $d$ the divisor function. This result is used to prove that the $L^q$ norm of $D_N(s):=\sum_{n=1}^{N} n^{-1/2-s}$ satisfies $\|D_N\|_q\gg (\log N)^{q/4}$ for $0<q<\infty$. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse inequality $\|D_N\|_q \ll (\log N)^{q/4}$ is shown to be valid in the range $1<q<\infty$. Similar bounds are found for a fairly large class of Dirichlet series including, on one of Selberg's conjectures, the Selberg class of $L$-functions.