Recursive formulas related to the summation of the Möbius function

Résumé

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.