The Hecke--Lerch Correspondence III: The Archimedean Completion and Gamma Factors

Part III of "The Hecke-Lerch Correspondence" SeriesContext:This is the third paper in a four-part series establishing the Hecke-Lerch Correspondence. Having established the discrete arithmetic theory for rational and quadratic fields in Parts I and II, this paper addresses the "place at infinity," extending the framework to the archimedean setting.Abstract:This paper extends the Hecke-Lerch correspondence to the archimedean setting, providing an explicit analytic construction of the Gamma factors that appear in completed L-functions.Building on discrete constructions for rational and imaginary quadratic fields, we define an archimedean Lerch transform by replacing summation with Fourier integration over a continuous spectral parameter. Using contour methods, we compute this transform in closed form, prove its nonvanishing, and show that its reciprocal kernel encodes the expected Gamma factor via Mellin transform. Specializing to $s=2, a=1/2$ confirms consistency with classical Lerch constants.We then assemble the local kernels into a global reciprocal object whose Mellin transform recovers completed L-functions through multiplicative factorization over all places, providing an explicit analytic alternative to the abstract adelic framework of Tate's thesis. This completes the local theory of the Lerch-reciprocal correspondence at all places of a global field.Series Links:Part I: The Rational Case and Explicit Class Field TheoryPart II: Imaginary Quadratic Fields and the Analytic Construction of Class Field TheoryPart IV: The Global Operator and the Riemann Hypothesis