The Hecke--Lerch Correspondence I: An Explicit Analytic Construction of Class Field Theory over Q via the LerchTranscendent

Part I of "The Hecke-Lerch Correspondence" SeriesContext:This is the first paper in a four-part series establishing the Hecke-Lerch Correspondence—an explicit analytic framework connecting the Lerch transcendent to the arithmetic of global fields. This paper lays the foundation by establishing the correspondence for the field of rational numbers $\mathbb{Q}$, providing an analytic realization of the Kronecker-Weber theorem.Abstract:This paper presents an explicit analytic framework for constructing the abelian extensions of the rational numbers using special functions, beginning from a family of elementary logarithmic integrals. By exploiting the structure of the Lerch transcendent and the rationality of the associated reciprocal generating functions, we produce a concrete analytic model of class field theory: every abelian extension of $\mathbb{Q}$ is encoded by suitable choices of parameters $(s,a)$.For rational parameters $a=r/k$, we prove that the coefficients of the reciprocal generating function decompose into Dirichlet characters modulo $k$. This allows the sequence to detect prime splitting in cyclotomic fields, leading to explicit denominator laws, analytic–arithmetic correspondences with Dirichlet L-functions, and a symmetry theory reflecting conductor conjugation. The work culminates in a general theorem giving an analytically explicit and computationally tractable realization of the Kronecker–Weber Theorem.Additionally, this framework proposes conjectural Hecke L-function identities and geometric chirality phenomena for imaginary quadratic fields, which are fully developed in Part II of this series. Extensive computational data supporting these findings is included.Series Links:Part II: Imaginary Quadratic Fields and the Analytic Construction of Class Field TheoryPart III: The Archimedean Completion and Gamma FactorsPart IV: The Global Operator and the Riemann Hypothesis