Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field

The aim of the present work is to exhibit a new proof of the explicit spectral expansion for the fourth moment of the Riemann zeta-function that was established by the second named author a decade ago. Our proof is new, particularly in the sense that it dispenses completely with the Kloostermania, the spectral theory of sums of Kloosterman sums that was used in the former proof. The argument is now constructed precisely upon the spectral structure of the Lie group PSL(2,R). M...

This is a rework of our old file, which has been left unpublished since September 1994, on an explicit spectral decomposition of the fourth power moment of the Riemann zeta-function against a weight which is the square of a Dirichlet polynomial. At this occasion we add an explicit treatment of generalized Kloosterman sums associated with arbitrary Hecke congruence subgroups (Section 15), which might have an independent interest. At the end (Section 36) of our discussion, we s...

We develop a reciprocity formula for a spectral sum over central values of L-functions on GL(4)xGL(2). As an application we show that for any self-dual cusp form Pi for SL(4,Z), there exists a Maass form pi for SL(2,Z) such that L(1/2, Pi x pi) is nonvanishing. An important ingredient is a "balanced" Voronoi summation formula involving Kloosterman sums on both sides, which can also be thought of as the functional equation of a certain double Dirichlet series involving Klooste...

In this paper we give an essential treatment for estimating arbitrary integral power moments of Kloosterman sums over the residue class ring. For prime moduli we derive explicit estimates, and for prime-power moduli we prove concrete formulas using computations with Igusa zeta functions.

The main aim of this article is to develop, in a fully detailed fashion, a {\bf unified} theory of the spectral theory of mean values of individual automorphic L-functions which is a natural extension of the fourth moment of the Riemann zeta-function but does not admit any analogous argument and requires a genuinely new method. Thus we first develop a relatively self-contained account of the theory of automorphic representations, especially highlighting the Kirillov model, wi...

We obtain several estimates for bilinear form with Kloosterman sums. Such results can be interpreted as a measure of cancellations amongst with parameters from short intervals. In particular, for certain ranges of parameters we improve some recent results of Blomer, Fouvry, Kowalski, Michel and Mili\'cevi\'c and also of Fouvry, Kowalski and Michel. In particular, we improve the bound on the error term in the asymptotic formula for mixed moments of $L$-series associated with H...

We derive explicit formulas for some Kloosterman sums on $\Gamma_0(N)$, and for the Fourier coefficients of Eisenstein series attached to arbitrary cusps, around a general Atkin-Lehner cusp.

We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.

In this paper we study asymptotic moment of fourth power twisted Kloosterman sum. We write this moment as weighted sums of Hurwitz class numbers and to obtain the asymptotic formula we use the theory of harmonic Maass forms and Mock modular forms. As a result we obtain asymptotic averages of hypergeometric functions.

We use recently obtained bounds for sums of Kloosterman sums to bound the sum $\sum_{-D\leq d\leq D} \int_{-D}^D |\zeta(1/2+it,\lambda^d)|^4| \sum_{0<|\mu|^2\leq M} A(\mu)\lambda^d((\mu)) |\mu|^{-2it}|^2 {\rm d}t$, where $\lambda^d$ is the groessencharacter satisfying $\lambda^d((\alpha)) = \lambda^d(\alpha{\Bbb Z}[i]) = (\alpha /|\alpha|)^{4d}$, for $0\neq\alpha\in{\Bbb Z}[i]$, and $\zeta(s,\lambda^d)$ is the Hecke zeta function that satisfies $\zeta(s,\lambda^d) =(1/4)\sum_...