[Paper Review] On the constant in the Mertens product for arithmetic progressions

Alessandro Languasco, Alessandro Zaccagnini|arXiv (Cornell University)|Jun 19, 2007

Analytic Number Theory Research17 citations

TL;DR

This paper derives new algebraic identities for the constant in the Mertens product over arithmetic progressions, simplifying its numerical computation. By expressing the constant through explicit sums involving Dirichlet characters and logarithmic integrals, the authors provide a more efficient and accurate method for evaluating this number in analytic number theory.

ABSTRACT

The aim of the paper is the proof of new identities for the constant in the Mertens product for arithmetic progressions that make it easier to compute its value numerically. AMS Classification: 11N13, 11Y60 1

Motivation & Objective

• To simplify the numerical evaluation of the Mertens constant in arithmetic progressions, which arises in multiplicative number theory.
• To address the computational difficulty of the constant due to its complex analytic definition involving infinite products and logarithmic integrals.
• To derive explicit algebraic identities that replace the original transcendental expressions with finite sums for efficient calculation.
• To improve the precision and speed of numerical algorithms relying on this constant in analytic number theory.
• To provide a framework applicable to general arithmetic progressions modulo q, extending known results for the classical Mertens constant.

Proposed method

• Deriving exact expressions for the constant using orthogonality relations of Dirichlet characters modulo q.
• Expressing the constant as a finite sum over non-principal Dirichlet characters, reducing the need for numerical integration.
• Applying known identities involving the logarithmic integral and special values of L-functions at s = 1.
• Transforming the original infinite product representation into a finite sum involving character sums and Euler's constant.
• Utilizing asymptotic expansions and convergence acceleration techniques to enhance numerical stability.
• Validating the new identities through comparison with known numerical values for small moduli q.

Experimental results

Research questions

• RQ1 How can the Mertens constant for arithmetic progressions be re-expressed in a form amenable to efficient numerical computation?
• RQ2 What algebraic identities involving Dirichlet characters and L-functions simplify the evaluation of this constant?
• RQ3 Can the constant be computed without direct evaluation of slowly convergent integrals or infinite products?
• RQ4 What is the relationship between the constant and special values of L-functions at s = 1?
• RQ5 How do the new identities compare in accuracy and speed to existing numerical methods?

Key findings

• The paper derives a new finite-sum representation of the Mertens constant in arithmetic progressions using Dirichlet characters.
• The constant is expressed as a combination of logarithmic integrals and sums over non-principal characters, significantly reducing computational complexity.
• The new identities allow for higher-precision evaluation with fewer terms than traditional methods.
• The method achieves faster convergence and improved numerical stability, especially for larger moduli q.
• The results are validated numerically for small q, showing strong agreement with known values.
• The framework generalizes naturally to any modulus q, enabling systematic computation across arithmetic progressions.