A Unifying Principle in the Theory of Modular Relations

The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function Z ( s ) satisfying a suitable functional equation, one can generalize it to Z f ( s ) in the form of an integral involving the Mellin transform F ( s ) of a certain suitable function f ( x ) and process it further as Z ˜ f ( s ) . Under the condition that F ( s ) is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for Z ˜ f ( s ) . Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion).