An inequality of Hardy–Littlewood type for Dirichlet polynomials

Abstract

The Lq norm of a Dirichlet polynomial F(s)=∑n=1Nann−s is defined as‖F‖q:=(limT→∞⁡1T∫0T|F(it)|qdt)1/q for 0<q<∞. It is shown that(∑n=1N|an|2|μ(n)|[d(n)]log⁡qlog⁡2−1)1/2≤‖F‖q when 0<q<2; here μ is the Möbius function and d the divisor function. This result is used to prove that the Lq norm of DN(s):=∑n=1Nn−1/2−s satisfies ‖DN‖q≫(log⁡N)q/4 for 0<q<∞. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse inequality ‖DN‖q≪(log⁡N)q/4 is shown to be valid in the range 1<q<∞. 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.