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Mertens function
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(Definition)
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Given an integer $n > 0$
$$M(n) = \sum_{i = 1}^n \mu(i)$$
where $\mu(x)$ is the Möbius function.
There is no $x$ such that $M(x) > x$ Franz Mertens conjectured that there is no $x$ such that $M(x) > \sqrt{x}$ but this was proven false later.
- 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)
Möbius and Mertens Values For n = 1 to 2500
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"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
There are 5 references to this entry.
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 2981 times total.
Classification:
| AMS MSC: | 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas) |
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Pending Errata and Addenda
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