Some Dirichlet series with coefficients related to periods of automorphic eigenforms, II

Abstract

6.This paper is a direct continuation of [2].Our primary objective here is to begin a discussion ot several applications of the general formalism considered in 2-5.7. We start by deriving some estimates for F(;S).Cf.Theorem 2. The basic procedure is that of analytic number theory.By examining appropriate combinations of the Mellin transforms mentioned in [2, p. 416 (line 5)] and applying (4.1), we quickly establish that (7.1)[Fp( S*)]:O(1)e /+)t for =a+it, aN, ]t]l, 30.The implied constant may depend on N, , .Compare [6, pp.311, 313] and [15, p. 22 (line 12)].We (can) now combine a Phragmn-LindelSf argument with (4.1)and theorem 2(v).Cf. [5, p. 95].This yields" Theorem 3. Given 01/100 and N3.Then" F(; S)=O[, t[ (o,/-,/+-)] for =a+it, [a]gN, [t]l.The implied constant depends solely on (F, N, S, ).8. Take T2x2000 and consider the integral

Locations

• Proceedings of the Japan Academy Series A Mathematical Sciences - View - PDF