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...

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...

We present a generalisation of the famous Selberg integral. This confirms the g = An case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for each simple Lie algebra g. Résumé. On présente une généralisation de la bien connue intégrale de Selberg. Cette généralisation vérifie le cas g = An de la conjecture de Mukhin et Varchenko concernant l’existence d’un...

In this paper we consider Selberg-type square matrices integrals with focus on Kummer-beta types I & II integrals. For generality of the results for real normed division algebras, the generalized matrix variate Kummer-beta types I & II are defined under the abstract algebra. Then Selberg-type integrals are calculated under orthogonal transformations.

in this paper we consider selberg-type square matrices integrals with focus on kummer-beta types i & ii integrals. for generality of the results for real normed division algebras, the generalized matrix variate kummer-beta types i & ii are defined under the abstract algebra. then selberg-type integrals are calculated under orthogonal transformations.

For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification po...

In this paper, some asymptotic formulas are proved for the harmonic mollified second moment of a family of Rankin-Selberg Lfunctions. One of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average. A first consequence is a new subconvexity bound for Rankin-Selberg L-functions...

It has been remarked that a fair measure of the impact of Atle Selberg’s work is the number of mathematical terms that bear his name. One of these is the Selberg integral, an n-dimensional generalization of the Euler beta integral. We trace its sudden rise to prominence, initiated by a question to Selberg from Enrico Bombieri, more than thirty years after its initial publication. In quick succe...

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.