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Shifted Multiple Dirichlet Series (1412.5917v1)

Published 18 Dec 2014 in math.NT

Abstract: We develop certain aspects of the theory of shifted multiple Dirichlet series and study their meromorphic continuations. These continuations are used to obtain explicit spectral first and second moments of Rankin-Selberg convolutions. One consequence is a Weyl type estimate for the Rankin-Selberg convolution of a holomorphic cusp form and a Maass form with spectral parameter $|t_j|\le T$, namely: $$ \left| L\left(\frac{1}{2}+ir, f\times u_j\right) \right| \ll_N T{2/3+\epsilon}, $$ uniformly, for $|r| \le T{2/3}$, with the implied constant depending only on $f$ and the level $N$.

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