Euler constant as a renormalized value of Riemann zeta function at its pole. Rationals related to Dirichlet L-functions

TL;DR

This paper redefines the value of the Riemann zeta function at 1 as the Euler constant, proposing a limit-based regularization and exploring its implications for number theory and related functions.

Contribution

It introduces a new limit definition for the zeta function at 1, linking Euler constant to a natural value, and explores rational sequences in Dirichlet L-functions with potential extensions of the Liouville lambda function.

Findings

Euler constant as a natural value of zeta at 1.

Asymptotic expansions involving Euler constant and zeta at odd integers.

Connections between rational sequences, Dirichlet L-functions, and the Liouville lambda function.

Abstract

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit definition of a function whose values coincide everywhere with those of the Riemann zeta function, save in 1, where our limit definition yields the Euler constant. Since in the literature one can find more than one way to regularize the value of the zeta function at s=1, we give asymptotic expansions where, by dint of some extended analogies, Euler constant appears to be the true "renormalized" value. As a striking example of such analogies, we propose an expansion of the logarithm function based on Euler constant and on all values of the zeta function at odd positive integers, in which all these presumably irrational numbers are accompanied by Harmonic numbers of corresponding orders. The other aim of this paper is to show how sequences of rationals, often the same, arise in computations related to Dirichlet L-functions. Here, a connection with the Liouville lambda function appears to have been found. Thus we raise the question about the possible usefulness of an extension of the Liouville lambda function to rationals. .