An inequality of Hardy-Littlewood type for Dirichlet polynomials

The L-q norm of a Dirichlet polynomial F(s) = Sigma(N)(n=1) a(n)n(-s) is defined as [GRAPHICS] for 0 < q < infinity. It is shown that [GRAPHICS] when 0 < q < 2; here mu is the Mobius function and d the divisor function. This result is used to prove that the L-q norm of D-N(s) := Sigma(N)(n=1) n(-1/2-s) satisfies parallel to D-N parallel to(q) >> (log N)(q/4) for 0 < q < infinity. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse inequality parallel to D-N parallel to(q) << (log N)(q/4) is shown to be valid in the range 1 < q < infinity. 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. (C) 2015 Elsevier Inc. All rights reserved.