Ramanujan’s Formula and Modular Forms

Abstract

In the theory of zeta-functions, which are defined wherever there are defined norms or substitutes thereof, the ingredients — modular relations, functional equations, incomplete gamma series, and the like — are placed like nodes on the woofs. Some of them are woven by warps as Hecke theory or Lavrik’s theory. The former connects the modular relation to the functional equation, thus making it possible to go to and from between the more orderly world of automorphic forms and the less orderly one of zeta-functions while the latter relates functional equations and incomplete gamma series in the same vein, the idea originating from Riemann. We have found a warp stitching all of these nodes-ingredients, enabling us to warp from one node to another as well as providing us with a guiding principle to locate the exact position and direction of research, a guiding thread to give a clear picture of the whole scene through opaque mist of complexity. We shall illustrate the principle by examples of various zeta-functions satisfying Hecke’s functional equation, i.e. the one with a single gamma factor, in which category many of the important zeta-functions are contained, notably, the Riemann zeta-, Dirichlet L-, Epstein zeta-, the automorphic zeta-functions, etc. In particular, we shall be concerned with the automorphic zeta-functions, the zeta functions arising from automorphic forms, evaluating their special values and obtaining incomplete gamma series.Keywordsfunctional equationHecke theoryincomplete gamma seriesmodular relationRamanujan’s formulaRiemann-Siegel integral formula