Exact Estimates for Double Dirichlet Series with Nonnegative Coefficients

Abstract

We consider the double Dirichlet series $$\sum\limits_{j = 2}^\infty {\sum\limits_{k = 2}^\infty {a_{jk} j^{ - 1 - x} k^{ - 1 - y} } } = : f(x,y) for x,y > 0,$$ with coefficients ajk ≧ 0 for all j, k ≧ 2. Among others, we give a necessary and sufficient condition in order that the above Dirichlet series converge for all x, y > 0; and prove exact estimates for certain weighted Lr-norms of f over the unit square (0,1) × (0,1) for any 0 < r < ∞, in terms of the coefficients ajk. Our approach is based on the exact estimates for integrals involving power series, from which we derive exact estimates for integrals involving Dirichlet series. All the results proved can be extended with ease to d-multiple Dirichlet series, where d is an arbitrary positive integer.