bibkey: languasco2008mertensprogressions authors: Alessandro Languasco; Alessandro Zaccagnini year: 2008 title: On the constant in the Mertens product for arithmetic progressions. I. Identities doi: null url: https://arxiv.org/pdf/0706.2807v4 claim: Introduction equation (1) recalls the fixed-modulus Mertens product in a reduced residue class; its stated asymptotic alone supplies no uniform moving-modulus bound. strata_touched: [] license: citation-only triage: anchor

Character slices and fixed versus varying moduli

This note records the authors’ arXiv version 0706.2807v4, dated September 26, 2008. On page 1, Introduction equation (1) recalls Williams’s theorem: for fixed coprime integers a,q, the product of 1-1/p over primes p <= x, p = a mod q is C(q,a)*(log x)^(-1/phi(q)) + O((log x)^(-1/phi(q)-1)), with positive C(q,a). The page also expresses the constants through Dirichlet L-functions; Section 2 on page 3 develops character identities.

A nonprincipal quadratic character takes value one on half of the unit classes. For a fixed modulus, combining those classes gives the classical square-root logarithmic growth of the corresponding reciprocal Euler product. This supplies context for FIB theory §§177–181, not an originality claim for the use of quadratic characters.

The source’s page 1 points to the authors’ separate 2007 paper for a result uniform in the modulus. That separate result’s full hypotheses have not been checked here and are not used. The project instead derives its explicit bound directly at real s > 1, with the modulus visible in log(m)+2. One cannot substitute a fixed-modulus asymptotic for that bound when the Lucas gap, discriminant and character vary with the integer source.