Explicit conversions between summatory functions of the Möbius function

We present an efficient explicit mechanism to convert bounds on $M (x) = \sum_{n \leq x} μ (n)$ to bounds on $m (x) = \sum_{n \leq x} μ (n)$ $∕$ $n$ and on $\overset{ˇ}{m} (x) = \sum_{n \leq x} (μ (n)$ $∕$ $n) \log (x$ $∕$ $n)$ . We use this mechanism in three different ways. We first improve on existing bounds for $m (x)$ and $\overset{ˇ}{m} (x)$ ; secondly, we compute the exact value of $\sup_{x \geq 1} (\log^{2} x)$ $|$ $\overset{ˇ}{m} (x) - 1$ $|$ and thirdly, we prove that $\lim^{¯}$ $|$ $m (x)$ $|$ $\sqrt{x} > \sqrt{2}$ . This establishes that the supremum of $| m (x) | \sqrt{x}$ is not reached for $x \to 2^{-}$ , contrary to what the first numerical observations might suggest.

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Keywords

explicit theory of prime numbers, asymptotics of arithmetical functions

Mathematical Subject Classification

Primary: 11N37

Milestones

Received: 11 October 2023

Revised: 19 April 2025

Accepted: 3 May 2025

Published: 22 May 2025

Authors

Florian Daval

Vol. 14, No. 2, 2025

Florian Daval

Abstract

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Keywords

Mathematical Subject Classification

Milestones

Authors