From explicit estimates for the primes to explicit estimates for the M\"obius function II

TL;DR

This paper enhances explicit estimates for the M"obius function by incorporating recent finite range computations, resulting in tighter bounds for sums involving the M"obius function for large X.

Contribution

It provides improved explicit bounds for the M"obius function sums by integrating recent computational data and refining existing methods.

Findings

Bound for |∑_{n≤X} μ(n)| with a smaller constant

Bound for |∑_{n≤X} μ(n)/n| with a smaller constant

Applicable for large X, starting from specific thresholds

Abstract

We improve on all the results of [13] by incorporating the finite range computations performed since then by several authors. Thus we have \begin{align*} \Bigg|\sum_{n\le X}\mu(n)\Bigg| &\le \frac{0.006688\,X}{\log X},&&\text{for } X\ge 1\,798\,118, \\\Bigg|\sum_{n\le X}\frac{\mu(n)}{n}\Bigg| & \le \frac{0.010032}{\log X},&& \text{for } X\ge 617\,990. \end{align*} We also improve on the method described in [13] by a simple remark.