It is known that theLp-norms of the sums of power series can be estimated from below and above by means of their coefficients, provided these coefficients
are nonnegative. In the paper we prove analogous estimates for theLp-norms of the sums of Dirichlet series. Our main result gives exact lower and upper estimates for the BMO-norm of the sums
of power series and Dirichlet series, respectively, by means of their coefficients.