An inequality of Hardy--Littlewood type for Dirichlet polynomials

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 \[ (\sum_{n=1}^{N} |a_n|^2|μ(n)|[d(n)]^{\frac{\log q}{\log 2} -1})^{1/2}\le \| F\|_q \] when $0<q<2$; here $\mu$ is the Möbius 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 $\Vert D_N{\Vert }_q\gg (\log N{)}^{q/4}$ for $0<q<\infty$. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse inequality $\Vert D_N{\Vert }_q\ll (\log N{)}^{q/4}$ 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.

arXiv commentThis is the final version of this paper, to appear in Journal of Number Theory

References in corpus (4)

• Continuous lower bounds for moments of zeta and L-functions
• Sharp conditional bounds for moments of the Riemann zeta function
• Integral means and boundary limits of Dirichlet series
• Moments of the Dedekind zeta function and other non-primitive L-functions

Cited by in corpus (12)

• Helson's problem for sums of a random multiplicative function
• Contractive inequalities for Bergman spaces and multiplicative Hankel forms
• Contractive inequalities for Hardy spaces
• Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function
• Volterra operators on Hardy spaces of Dirichlet series
• Composition operators and embedding theorems for some function spaces of Dirichlet series
• Pseudomoments of the Riemann zeta function
• Contractive inequalities for mixed norm spaces and the Beta function
• Convergence Abscissas for Dirichlet Series with Multiplicative Coefficients
• A contractive Hardy-Littlewood inequality
• High pseudomoments of the Riemann zeta function
• A Sharp Inequality of Hardy-Littlewood Type Via Derivatives