An overview of the classical Rankin-Selberg problem involving the asymptotic formula for sums of coefficients of holomorphic cusp forms is given. We also study the function ∆(x; ξ) (0 ≤ ξ ≤ 1), the error term in the Rankin-Selberg problem weighted by ξ-th power of the logarithm. Mean square estimates for ∆(x; ξ) are proved. 1. The Rankin-Selberg problem The classical Rankin-Selberg problem cons...

In this paper, new examples of weakly symmetric spaces in the sense of A. Selberg 14] are constructed. Let G be a connected, simply-connected, simple Lie group of hermitian type and K a maximal compactly embedded subgroup of G such that D = G=K is an irreducible classical bounded symmetric domain. Let G 1 and D 1 be circle extensions of G and D, respectively. The factor of automorphy induces a ...

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.

In this article, we study p-adic torus periods for certain p-adic valued functions on Shimura curves coming from classical origin. We prove a p-adic Waldspurger formula for these periods, as a generalization of the recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic p-adic L-function of Rankin– Selberg type. At a character of positive w...

In this paper we consider two Rankin-Selberg integrals which were discovered by David Ginzburg, and announced in [G-H1]. These integrals are defined on a split form of GSO2n (see below for precise definition), and involve a generic cuspidal automorphic representation of this group. We content ourselves with showing that both integrals unfold to Eulerian integrals involving Whittaker functions, ...

α∈R (∣∣〈ρk, α∨〉+ kα + 1 2kα/2∣∣)! (∣∣〈ρk, α∨〉+ 1 2kα/2∣∣)! • Also in 1982, A. Koranyi uses the Selberg formula to compute the volumes of bounded symmetric domains. • In 1987, K. Aomoto studies a slight generalization of the Selberg integral arising from work on Fock space representations of the Virasoro algebra. Here a connection with hypergeometric functions, specifically Jacobi polynomials is...

The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects “Young books” are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases...