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arXiv · 2408.05969

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

Olivier Ramaré·Sebastian Zuniga Alterman
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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.

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Olivier Ramaré, Sebastian Zuniga Alterman. 2024-08-12. From explicit estimates for the primes to explicit estimates for the M\"obius function II. https://doi.org/10.7169/facm%2F250121-19-5

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