Published February 3, 2026 | Version version 1
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Explicit Analytic Formulas for Zeros of Generalized L-Functions with Rigorous Derivations

Authors/Creators

  • 1. ROR icon Peking University

Description

This paper establishes a unified theoretical framework for complete explicit analytic formulas of nontrivial zeros of Selberg class and more generalized L-functions. Based on the kernel function method with adjustable parameters, we construct rapidly convergent summation formulas for generalized L-functions and rigorously prove the super-exponential convergence of the series. On this basis, we derive explicit analytic equations for the ordinates of zeros, expressing zeros as combinations of rapidly convergent series, Gamma functions, and Digamma functions, completely eliminating reliance on implicit summation over zeros. This framework applies to Dirichlet L-functions, Hecke L-functions, automorphic L-functions, Artin L-functions, as well as broad classes lacking Euler products such as Hurwitz zeta functions. We systematically optimize constants in the formulas, propose efficient parallel iterative algorithms, reducing computational complexity from O(T2/3+ϵ) of traditional methods to O(T1/2+ϵ). As applications, we improve zero-density estimates, establish precise correspondences with random matrix theory, and reveal profound connections between explicit formula structures and periodic orbit theory in quantum chaos,providing new tools for interdisciplinary research in analytic number theory and mathematical physics.This paper further rigorously extends the theoretical framework to operator-valued L-functions, establishes compatibility with Motivic L-functions and the Langlands program, ultimately constructing a grand unified theory concerning L-function zeros, transforming related conjectures into rigorous theorems.

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Alternative title (English)
Explicit Analytic Formulas for Zeros of Generalized L-Functions with Rigorous Derivations

Dates

Submitted
2025-12-31

References

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