Multiple Dirichlet series of several complex variables are considered. Using the Mellin-Barnes integral formula, we prove the analytic continuation and an upper bound estimate.
We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if with , then and even slightly better, and , C being an absolute constant.
For a number field with ring of integers , we prove an analogue over finite rings of the form of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where is a big enough prime ideal of and . In the appendix, F.Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J.Denef and A.Gyoja [Character sums associated to prehomogeneous vector spaces, Compos. Math., 113(1998), 237–346] to the functional equation of -functions...
We study the second moment of the central values of quadratic twists of a modular -function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.