An Analogue of the Chowla–selberg Formula for Several Automorphic L-functions
نویسنده
چکیده
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 consisting of K-Bessel functions. In addition, we also give certain analogues of the Chowla-Selberg formula for Dirichlet L-functions and L-functions associated with holomorphic cusp forms. Moreover, we introduce a two variable function which is analogous to the real analytic Eisenstein series and give a certain limit formula for this one. Such a limit formula can be regarded as an analogue of Kronecker’s limit formula.
منابع مشابه
The Chowla-Selberg Formula for Quartic Abelian CM Fields
We provide explicit analogues of the Chowla-Selberg formula for quartic abelian CM fields. This consists of two main parts. First, we implement an algorithm to compute the CM points at which we will evaluate a certain Hilbert modular function. Second, we exhibit families of quartic fields for which we can determine the precise form of the analogue of the product of gamma values.
متن کاملScattering Theory for Automorphic Functions
This paper is an expository account of our 1976 monograph [6] on Scattering theory for automorphic functions. Several improvements have been incorporated: a more direct proof of the meromorphic character of the Eisenstein series, an explicit formula for the translation representations and a simpler derivation of the spectral representations. Our hyperbolic approach to the Selberg trace formula ...
متن کاملMultidimensional extension of the generalized Chowla–Selberg formula
After recalling the precise existence conditions of the zeta function of a pseudodifferential operator, and the concept of reflection formula, an exponentially convergent expression for the analytic continuation of a multidimensional inhomogeneous Epstein-type zeta function of the general form ζA,~b,q(s) = ∑ ~n∈Z (~nA~n+~b~n+ q)−s, with A the p× p matrix of a quadratic form, ~b a p vector and q...
متن کاملSpecial values of multiple gamma functions par William DUKE
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.
متن کاملArithmetic Theta Lifts and the Arithmetic Gan–gross–prasad Conjecture for Unitary Groups
In 1980s, Gross–Zagier [GZ86] established a formula that relates the Neron–Tate height of Heegner points on modular curves to the central derivative of certain L-functions associated to modular forms. Around the same time, Waldspurger proved a formula, relating toric periods of modular forms to the central value of certain L-functions. Gross put both of these formula in the framework of represe...
متن کامل