Further insights into the mysteries of the values of zeta functions at integers

TL;DR

This paper offers a simple interpretation of zeta function values at negative integers and zero, linking them to areas of partial sums of powers, and explores implications for L-functions and ongoing research.

Contribution

It introduces a natural interpretation of zeta values at negative integers, connecting them to partial sum areas, and extends these ideas to L-functions, providing new insights.

Findings

Relates zeta values at negative integers to areas of partial sums

Provides a new interpretation of zeta function values at zero

Suggests potential connections with broader research on zeta functions

Abstract

We present a remarkably simple and surprisingly natural interpretation of the values of zeta functions at negative integers and zero. Namely we are able to relate these values to areas related to partial sums of powers. We apply these results to further interpretations of values of $L$-functions at negative integers. We hint in a very brief way at some expected connections of this work with other current efforts to understand the mysteries of the values of zeta functions at integers.