Recursive formulas related to the summation of the Möbius function

  1. Benito Muñoz, Manuel
  2. Varona Malumbres, Juan Luis
Journal:
Open Mathematics Journal

ISSN: 1874-1177

Year of publication: 2008

Volume: 1

Pages: 25-34

Type: Article

DOI: 10.2174/1874117400801010025 GOOGLE SCHOLAR lock_openOpen access editor

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Abstract

Let M(x)=∑⌊x⌋k=1μ(k) be the Mertens function, where μ is the Möbius function. The Mertens conjecture, that |M(x)|<x√, was disproved indirectly by A. M. Odlyzko and H. J. J. te Riele [J. Reine Angew. Math. 357 (1985), 138–160; MR0783538 (86m:11070)]. The authors present several formulas and identities for M(x) with the aim of aiding the search for an explicit counterexample to the Mertens conjecture. The paper's main goal is to reduce the number of summands in the calculation of the Mertens function. To this end, the authors use previously known identities for M(x) and combinatorial arguments to give a two-parameter formula (Theorem 10) for calculating M(x) involving previously known values and an additional term depending mainly on residues. No explicit calculations are made, but several specialized versions of this formula are derived; the formulas involved require no more than x/3 summands, depending on the number of previously calculated values.