Dirichlet eta and beta functions at negative integer arguments: Exact results from anti-limits

TL;DR

This paper introduces a novel method to evaluate Dirichlet eta and beta functions at negative integers using polynomial extrapolations, providing exact results and a new perspective on summability.

Contribution

It presents a direct, independent, and computationally efficient approach to evaluate divergent alternating sums at negative integers, generalizing previous lattice sum methods.

Findings

Exact evaluations of Dirichlet eta and beta functions at negative integers.

A new interpretation of summability for divergent series.

Enhanced understanding of anti-limits in mathematical analysis.

Abstract

A route to evaluate exact sums represented by Dirichlet eta and beta functions, both of which are alternating and divergent at negative integer arguments, is advocated. It rests on precise polynomial extrapolations and stands as a generalization of an early endeavor on lattice sums. Apart from conferring a physical meaning to anti-limits, the scheme advanced here is direct, independent and computationally appealing. A new interpretation of summability is also gained.