On Koshlyakov’s Transforms and Their Applications

Koshlyakov’s 1954 paper is an opus magnum containing almost all (a.a.) buds of modular relations, equivalent assertions to the functional equation of Dedekind zeta-functions of the rational and quadratic fields. We shall restore this in the framework of Koshlyakov–Oberhettinger–Soni consisting of zeta-functions corresponding to the gamma factor Γs2 (Riemann), Γ(s) (Hecke) and Γs22 (Voronoĭ-Bochner) and locate more recent developments in their proper position in the framework. In partic-ular, we elucidate the situation where Ramanujan’s kernels, the sinus cardinalis function, Poisson and Plana summation formulas, etc., take place, thus giving a clue for them to be modular relations.