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.
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