In this paper, we will give a certain formula for the Riemann zeta function that expresses the Riemann zeta function by an infinte series consisting of KBessel functions. Such an infinite series expression can be regarded as an analogue of the Chowla-Selberg formula. Roughly speaking, the Chowla-Selberg formula is the formula that expresses the Epstein zeta-function by an infinite series consis...

We study the asymptotic behavior of zeros of the Selberg zeta-function for the congruence subgroup Γ0(4) as a function of a one-parameter family of characters tending to the trivial character. The motivation for the study comes from observations based on numerical computations. Some of the observed phenomena lead to precise theorems that we prove and compare with the original numerical results....

A simple integral formula as an iterated residue is presented for the Baker-Akhiezer function related to An type root system both in the rational and trigonometric cases. We present also a formula for the Baker-Akhiezer function as a Selberg-type integral and generalise it to the deformed An,1case. These formulas can be interpreted as new cases of explicit evaluation of Selberg-type integrals.

More general results, including arbitrary Fuchsian groups, can be found in the paper [Bo2] of Borcherds, Sect. 7. Most of them have been proved by the Selberg trace formula, see [Iv] and also [Fi]. The Selberg trace formula in its standard form causes the restriction that the weight is > 2. Borcherds mentions that “with a bit more care this also works for weight 2”. As we mentioned, this bit mo...

We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [20]. The main tool is the Selberg trace formula which, unlike previous geometric methods, allows for treatment of cases where the eigenvalue 1 4 is present. We present a few other sample applications, including the classification of even 2-dimensional Galois represen...

We propose a geometric interpretation of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program. Our main result is local, its classical counterpart is the RankinSelberg computation of convolution of two Hecke eigenforms with an Eisenstein series. For n = 2 we apply it to prove a global result that gives a geometric version of the computation of the sc...

In this paper, a heuristic method to compute the Selberg zeta function for Hecke triangle groups, Gq is described. The algorithm is based on the transfer operator method and an overview of the relevant background is given.We give numerical support for the claim that the method works and can be used to compute the Selberg Zeta function on Gq to any desired precision. We also present some numeric...

We study eta invariants of Dirac operators and regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner Müller (see [18], [19]) and use relative traces to define the eta function and the zeta function. We show regularities of eta and zeta functions at the origin so that we can define the eta invariant and the regularized determinant. By the Selberg tra...

In [7] A. Selberg axiomatized properties expected of L-functions and introduced the “Selberg class” which is expected to coincide with the class of all arithmetically interesting L-functions. We recall that an element F of the Selberg class S satisfies the following axioms. • In the half-plane σ > 1 the function F (s) is given by a Dirichlet series ∑∞n=1 aF (n)n with aF (1) = 1 and aF (n) ≪ǫ n ...