Residue of a Mod 5 Euler Product

TL;DR

This paper evaluates the residue at s=1 of a specific Euler product over primes congruent to 1 mod 5, compares it with a known Mertens constant, and studies the distribution of primitive quintic Dirichlet characters.

Contribution

It provides a new evaluation of the residue of a specialized Euler product and analyzes the asymptotic count of primitive quintic Dirichlet characters.

Findings

Calculated the residue at s=1 of the Euler product.

Compared the residue with the Mertens constant of Languasco & Zaccagnini.

Determined the average number of primitive quintic Dirichlet characters as n approaches infinity.

Abstract

Consider the product of (1-p^(-s))^(-4) over all primes p=1 mod 5. We evaluate its residue at s=1 and compare with the corresponding Mertens constant of Languasco & Zaccagnini. We also count primitive quintic Dirichlet characters mod n and determine their average number as n->infty.