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...

All known ways to analytically continue automorphic L-functions involve integral representations using the corresponding automorphic forms. The simplest cases, extending Hecke’s treatment of GL2, need no further analytic devices and very little manipulation beyond Fourier-Whittaker expansions. [1] Poisson summation is a sufficient device for several accessible classes of examples, as in Riemann...

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...

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

We present the simplest possible example of the Rankin-Selberg method, namely for a pair of holomorphic modular forms for SL(2,Z), treated independently in 1939 by Rankin and 1940 by Selberg. (Rankin has remarked that the general idea came from his advisor and mentor, Ingham.) We also recall a proof of the analytic continuation of the relevant Eisenstein series. That is, we consider the simples...

Such identities arise in Rankin-Selberg integral representations of L-functions. For GL2, a naive, direct computation is sufficient. However, for general GLn and other higher-rank groups, direct computation is inadequate. Further, connecting local Rankin-Selberg computations to Schur functions usefully connects these computations to the Shintani-Casselman-Shalika formulas for spherical p-adic W...

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.