A Moebius sum

Olivier Ramar\'e

arXiv:2308.07632·math.NT·August 16, 2023

A Moebius sum

Olivier Ramar\'e

PDF

Open Access

TL;DR

This paper establishes numerical bounds for a specific sum involving the Möbius function and least common multiples, providing precise inequalities valid for all sufficiently large X.

Contribution

It introduces new bounds for the sum (X) involving the Möbius function and lcm, advancing understanding of its behavior.

Findings

01

0 (X) 17/25 for all X 2

02

Derived explicit numerical bounds for the sum involving (X)

03

Confirmed the sum is non-negative for all X 2

Abstract

We provide numerical bounds for $Σ (X) = \sum_{d_{1}, d_{2} \leq X} \frac{μ ( d _{1} ) μ ( d _{2} )}{[ d _{1} , d _{2} ]}$ . We show in particular that $0 \leq Σ (X) \leq 17/25$ for every $X \geq 2$ .

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Taxonomy

TopicsAnalytic Number Theory Research · Mathematical Approximation and Integration