Applications of the betatransform
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2010 Mathematics Subject Classification:

11R42, 11R11.

Keywords:

betatransform, Dedekind zetafunction, Hurwitz zetafunction, Kelvin function, quadratic fields.

Abstract:

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

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Vol. 10 (18), 2015
