Unified Theory of Zeta-Functions Allied to Epstein Zeta-Functions and Associated with Maass Forms

The Riemann Zeta and Allied Functions

Calculation and Applications of Epstein Zeta Functions

Rapidly convergent series are given for computing Epstein zeta functions at integer arguments. From these one may rapidly and accurately compute Dirichlet L functions and Dedekind zeta functions for quadratic and cubic fields of any negative discriminant. Tables of such functions computed in this way are described and numerous applications are given, including the evaluation of very slowly conv...

Higher moments of the Epstein zeta functions

holds as T → ∞. The basic tools of them are the approximate functional equations for ζ(s) and ζ(s). Therefore, one might think that we can obtain the asymptotic formula for the higher moments (sixth moment, eighth moment, etc· · · ) of ζ(s) on the critical line Re(s) = 12 by using the approximate functional equations for ζ(s) (k ≥ 3). However, although these approximate functional equations are...

Automorphic Forms and Related Zeta Functions

For a division quaternion algebra D with the discriminant two, we construct Maass cusp forms on GL2(D) explicitly by some lifting from Maass cusp forms for Γ0(2). This is an analogue of the Saito-Kurokawa lifting. We expect that this lifting leads to a construction of a CAP representation of GL2(D), which is an inner form of GL(4). The aim of the talk is to report recent progress of our study o...

Automorphic forms, trace formulas and zeta functions

Around 1980, K. Doi and H. Hida found a meaning of the special value of certain degree 3 L-functions, so called adjoint L-functions of cusp forms. They discovered that if a prime divides “algebraic part” of the adjoint L-function of a cusp form, the prime is a congruence prime for the cusp form. E. Ghate and M. Dimitrov proved analogues of Hida’s theorem in Hilbert modular case. E. Urban also p...