Applications of the beta-transform

Author:

Kalyan~Chakraborty,~Shigeru~Kanemitsu,~Haruo~Tsukada

2010 Mathematics Subject Classification:

11R42, 11R11.

Keywords:

beta-transform, Dedekind zeta-function, Hurwitz zeta-function, Kelvin function, quadratic fields.

Abstract:

The importance of the Hecke gamma-transform in number theory cannot be overstated. Along with this, there has been equally effective applications of the method of beta-transform, or the binomial expansion, leading to the K-Bessel series. The purpose here is to elucidate the explicit and implicit use of the beta-transform in the transformation of the perturbed Dirichlet series and reveal the hidden structure as the Fourier-Bessel expansion. In the first instance, we shall deal with Stark's method along with a recent result of Murty-Sinha as a manifestation of the beta-transform. Later and in the main part, we shall resurrect the long-forgotten important work of Koshlyakov [10, 11, 12, 13] showing that Koshlyakov's formula for the perturbed Dedekind zeta-functions for a real quadratic field is in fact Lipschitz summation formula, and that for an imaginary quadratic field the K-Bessel function is intrinsic to the Hecke functional equation. Similarly, we shall elucidate Koshlyalov's \sigma-series in terms of Kelvin functions.

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Vol. 10 (18), 2015