PlanetMath: Mertens function

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Mertens function

(Definition)

Given an integer ,

$\displaystyle M(n) = \sum_{i = 1}^n \mu(i)$

where $\mu(x)$ is the Möbius function.

There is no such that . Franz Mertens conjectured that there is no such that $M(x) > \sqrt{x}$ , but this was proven false later.

Bibliography

1: F. Mertens, ``Über eine zahlentheoretische Funktion" Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, IIa 106, 761-830 (1897)
2: A. M. Odlzyko and te Riele, ``Disproof of the Mertens Conjecture" Journal für die reine und angewandte Mathematik, 357, 138-160 (1985)

"Mertens function" is owned by CompositeFan.

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Other names:

Mertens' function, Mertens's function

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Cross-references: Möbius function, integer
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This is version 6 of Mertens function, born on 2006-04-15, modified 2006-06-13.
Object id is 7836, canonical name is MertensFunction.
Accessed 1596 times total.

Classification:

11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)

Pending Errata and Addenda

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