On Validated Numeric of Values of Some Zeta and L-functions, and Applications

TL;DR

The paper tackles the challenge of obtaining rigorous numeric values for $\zeta(s)$, Dirichlet $L$-functions, Dedekind zeta functions, and Hasse-Weil $L$-functions. It develops interval-based validated numerics, combining Euler products, Dirichlet series, and modularity-linked representations, with explicit error bounds. Key contributions include interval bounds for $\zeta(s)$ at $Re(s) \ge 1$, a concrete formula for $L(1,\chi)$ with bounded remainder, and a description of how modularity yields computable $L(E,1)$ values. These results enable rigorous numerical verification of critical values and support applications in arithmetic geometry, BSD-type conjectures, and related computational number theory.

Abstract

This paper describes some validated numerics aspects of Riemann zeta function, Dirichlet L-functions, Dedekind zeta functions and Hasse-Weil L-functions.