An inequality of Hardy--Littlewood type for Dirichlet polynomials

The $L^q$ norm of a Dirichlet polynomial $F(s)={\sum }_{n=1}^N{a}_n{n}^{-s}$ is defined as $$\mathrm{\parallel }F{\mathrm{\parallel }}_q:=(\underset{T\to \mathrm{\infty }}{\lim }\frac{1}{T}{\int }_0^T\mathrm{\mid }F(it){\mathrm{\mid }}^{q}dt{)}^{1\mathrm{/}q}$$ for $0<q<\mathrm{\infty }$. It is shown that $$(\sum _{n=1}^N\mathrm{\mid }{a}_n{\mathrm{\mid }}^2\mathrm{\mid }\mu (n)\mathrm{\mid }[d(n){]}^{\frac{\log q}{\log 2}-1}{)}^{1\mathrm{/}2}\le \mathrm{\parallel }F{\mathrm{\parallel }}_q$$ 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 $\mathrm{\parallel }{D}_N{\mathrm{\parallel }}_q\ll (\log N{)}^{q\mathrm{/}4}$ is shown to be valid in the range $1<q<\mathrm{\infty }$. 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.