On counterexamples to the Mertens conjecture

TL;DR

This paper employs advanced lattice algorithms to significantly lower the upper bound on the smallest counterexample to the Mertens conjecture, providing new computational evidence related to this longstanding mathematical question.

Contribution

It introduces improved lattice algorithms that reduce the upper bound on the first counterexample to the Mertens conjecture by several orders of magnitude.

Findings

Upper bound on the counterexample is approximately exp(1.96×10^{19})

Significantly below the previous conjectured bound of exp(5.15×10^{23})

Demonstrates the effectiveness of advanced lattice algorithms in number theory problems

Abstract

We use state-of-art lattice algorithms to improve the upper bound on the lowest counterexample to the Mertens conjecture to $\approx \exp(1.96 \times 10^{19})$, which is significantly below the conjectured value of $\approx \exp(5.15 \times 10^{23})$ by Kotnik and van de Lune [KvdL04].