Two elementary formulae and some complicated properties for Mertens function

2016/12/05 by Rong Wei, Wei, Rong Qiang
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Matrix Theory and Algorithms #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1612.01394

openalex publication_date 2016/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Two elementary formulae for Mertens function M(n) are obtained. With these formulae, M(n) can be calculated directly and simply, which can be easily implemented by computer. M (1) ∼ M (2 × 10⁷) are calculated one by one. Based on these 2× 10⁷ samples, some of the complicated properties for Mertens function M(n), its 16479 zeros, the 10043 local maximum/minimum between two neighbor zeros, and the relation with the cumulative sum of the squarefree integers, are understood numerically and empirically.

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