From explicit estimates for the primes to explicit estimates for the Möbius function II

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}μ(n)\Bigg| &\le \frac{0.006688,X}{\log X},&&\text{for } X\ge 1,798,118, \\Bigg|\sum_{n\le X}\frac{μ(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.

💡 Research Summary

The paper “From explicit estimates for the primes to explicit estimates for the Möbius function II” by Olivier Ramaré and Sebastián Zúñiga‑Al Termán revisits the problem of obtaining explicit, numerically effective bounds for the summatory Möbius function
(M(X)=\sum_{n\le X}\mu(n)) and for its logarithmic average (\displaystyle m(X)=\sum_{n\le X}\frac{\mu(n)}{n}).
The authors build directly on their earlier work