A new upper bound on the smallest counterexample to the Mertens conjecture

Abstract

We report the finding of the new upper bound on the lowest positive integer $x$ for which the Mertens conjecture \begin{equation*} \left| \sum_{1 \leq n \leq x} μ(n) \right| < \sqrt{x} \end{equation*} fails to hold: $x<\exp (1.017\times 10^{29})$, an improvement over previously known $\exp (1.59\times 10^{40})$ due to Kotnik and te Riele [7].