Improvements to Turing's Method

TL;DR

This paper enhances Turing's method by refining Lehman's argument, reducing constants, and extending improvements to related L-functions and zeta-functions, thereby advancing computational techniques in analytic number theory.

Contribution

It provides a refined version of Turing's method with smaller constants and extends these improvements to Dirichlet L-functions and Dedekind zeta-functions.

Findings

Reduced constants in Turing's method as per Theorem 1

Extended improvements to Dirichlet L-functions

Extended improvements to Dedekind zeta-functions

Abstract

This paper refines the argument of Lehman by reducing the size of the constants in Turing's method. This improvement is given in Theorem 1 and scope for further improvements is also given. Analogous improvements to Dirichlet L-functions and Dedekind zeta-functions are also included.