The paper gives complete proofs of the properties of the RankinSelberg integrals for the group GL(n,R) and GL(n,C).

Abstract We propose a version of the Selberg trace formula for compact hyperbolic orbifolds $$\Gamma \backslash {\mathbb {H}}^{2n+1}$$ Γ \ H 2 n + 1 non-unitary r...

In the present paper, we study growth of Selberg zeta function for modular group in critical strip.

The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces cusp forms in terms weighted sums Hurwitz-Kronecker class numbers. For $\text {\rm SL}_2(\mathbb{Z}),$ Zagier proved these by cleverly making use weight 3/2 nonholomorphic Eisenstein series he discovered 1970s. holomorphic part this form, its so-called {\it mock modular form}, is generating function for In expo...

In this paper, we solve the Rankin--Selberg problem. That is, break well known Rankin--Selberg's bound on error term of second moment Fourier coefficients a $\mathrm{GL}(2)$ cusp form (both holomorphic and Maass), which remains its record since birth for more than 80 years. We extend our method to deal with averages L-functions can be factorized as product degree one three L-functions.

We give a Chowla-Selberg type formula that connects a generalization of the eta-function to GL(n) with multiple gamma functions. We also present some simple infinite product identities for certain special values of the multiple gamma function.

We describe two new Eulerian Rankin-Selberg integrals, using the same Eisenstein series defined on the group E8, and cuspidal representations from GL5 and GSpin11, respectively. Connections with past work of Ginzburg, Bump-Ginzburg, Jiang-Rallis and others are described. We give some details of how to relate our two integrals via formal manipulations. It is a fact of classical invariant theory ...