A discrete approach to Dirichlet L-functions, their special values and zeros

We develop a discrete spectral framework for Dirichlet $L$-functions that reveals a combinatorial structure underlying their special values and connects this to their zeros. Our approach approximates the classical Dirichlet series by finite spectral sums $L_n(s,\chi)$ associated with cyclic graphs $\mathbb{Z}/n\mathbb{Z}$ and studies their asymptotics as $n\rightarrow \infty$. Combining a refined Euler Maclaurin expansion with a structural polynomiality property, we show that at integer arguments the asymptotic expansions terminate and yield exact identities. This asymptotic to exact principle produces new infinite families of relations among special values of Dirichlet $L$-functions and recovers, by a different mechanism, formulas previously obtained by Xie, Zhao and Zhao. An interesting feature of our method is that $\zeta(2n)$ and the corresponding special values for all Dirichlet $L$-functions thereby admit a finite combinatorial interpretation in terms of rooted spanning forests on any fixed cyclic graph. Concerning zeros, the same framework leads to some remarks about real zeros and a reformulation of the Generalized Riemann Hypothesis in the case of odd primitive characters in terms of an asymptotic functional equation relating $\xi_n(1-s,\overline{\chi})$ to $\xi_n(s,\chi)$ of the completed discrete functions. This establishes the remaining case of the one dimensional picture obtained in earlier works.