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A discrete approach to Dirichlet L-functions via spectral models: special values and zeros

Anders Karlsson, Dylan Müller

TL;DR

We reinterpret Dirichlet L-functions as spectral invariants of Laplacians and introduce finite discrete analogs L_n from cyclic graphs to approximate them. The paper develops exact special-value formulas and zeros results using a refined Euler-Maclaurin-Sidi expansion together with a polynomiality property of spectral sums. The work also establishes an equivalence between GRH for odd primitive Dirichlet characters and a discrete asymptotic functional equation, and analyzes real zeros via heat-positivity criteria with explicit examples. Overall, the results fuse discrete graph spectral methods with classical analytic number theory to yield new trig-sum identities, recursion relations for zeta and L-values, and insights into GRH from a finite-graph perspective.

Abstract

We study Dirichlet $L$-functions via discrete analogs $L_n$ arising from the spectral theory of cyclic graphs as $n\rightarrow \infty$. Using a refined Euler-Maclaurin asymptotic expansion due to Sidi, together with an independent polynomiality property of these finite spectral sums at integers, we obtain exact special-value formulas, even starting at $n=1$. This yields new expressions for certain trigonometric sums of interest in physics, and recovers, by a different method, the striking formulas of Xie, Zhao, and Zhao. It also gives infinite families of recursion relations among special values of the Riemann zeta function and of Dirichlet $L$-functions. Concerning zeros, we prove that, for odd primitive characters, a natural asymptotic functional equation for the discrete functions $L_n$ is equivalent to the Generalized Riemann Hypothesis for the corresponding Dirichlet $L$-function. We also provide some remarks about the non-existence of possible real zeros.

A discrete approach to Dirichlet L-functions via spectral models: special values and zeros

TL;DR

We reinterpret Dirichlet L-functions as spectral invariants of Laplacians and introduce finite discrete analogs L_n from cyclic graphs to approximate them. The paper develops exact special-value formulas and zeros results using a refined Euler-Maclaurin-Sidi expansion together with a polynomiality property of spectral sums. The work also establishes an equivalence between GRH for odd primitive Dirichlet characters and a discrete asymptotic functional equation, and analyzes real zeros via heat-positivity criteria with explicit examples. Overall, the results fuse discrete graph spectral methods with classical analytic number theory to yield new trig-sum identities, recursion relations for zeta and L-values, and insights into GRH from a finite-graph perspective.

Abstract

We study Dirichlet -functions via discrete analogs arising from the spectral theory of cyclic graphs as . Using a refined Euler-Maclaurin asymptotic expansion due to Sidi, together with an independent polynomiality property of these finite spectral sums at integers, we obtain exact special-value formulas, even starting at . This yields new expressions for certain trigonometric sums of interest in physics, and recovers, by a different method, the striking formulas of Xie, Zhao, and Zhao. It also gives infinite families of recursion relations among special values of the Riemann zeta function and of Dirichlet -functions. Concerning zeros, we prove that, for odd primitive characters, a natural asymptotic functional equation for the discrete functions is equivalent to the Generalized Riemann Hypothesis for the corresponding Dirichlet -function. We also provide some remarks about the non-existence of possible real zeros.

Paper Structure

This paper contains 18 sections, 22 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Bundle Laplacian and asymptotics
  3. Hurwitz zeta functions
  4. Bundle Laplacian
  5. A unitary line bundle over the discrete circle
  6. The continuous circle
  7. Asymptotics
  8. Asymptotics of discrete Dirichlet $L$-functions
  9. Equivalence with GRH for odd character.
  10. Real zeros of Dirichlet $L$-functions
  11. No real zero for Dirichlet $L$-functions with heat positive character
  12. First examples of heat positive characters
  13. Further examples of heat positive characters
  14. Special values
  15. Formulas
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\chi$ be an odd Dirichlet character of modulus $q > 1$. Then, for any fixed $s \in \mathbb{C}$ and any integer $m \ge 0$, the following asymptotic expansion holds as $n \to \infty$: where the functions $b_k(s)$ are polynomials of degree $k$, given by the Laurent expansion

Theorems & Definitions (39)

  • Theorem 1.1: Asymptotics for odd characters
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5: of Theorem \ref{['Asymptotic for L tilde']}
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 2.1
  • Remark 2.2
  • ...and 29 more