Mertens function

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In number theory, the Mertens function is

where μ(k) is the Möbius function. The function is named in honour of Franz Mertens.

Because the Möbius function has only the return values -1, 0 and +1, it's obvious that the Mertens function moves slowly and that there is no x such that M(x) > x. The Mertens conjecture goes even further, stating that there is no x where the absolute value of the Mertens function exceeds the square root of x. The Mertens conjecture was disproven in 1985. However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(x), namely <math>M(x) = o(x^{\frac12 + \epsilon})</math>. Since high values for M grow at least as fast as the square root of x, this puts a rather tight bound on its rate of growth. Here, <math>o</math> refers to little-o notation.

[edit] Integral representations

Using the Euler product one finds that

<math> \frac{1}{\zeta(s) }= \prod_{p} (1-p^{-s})= \sum_{n=1}^{\infty}\mu (n)n^{-s} </math>

where <math>\zeta(s)</math> is the Riemann zeta function and the product is taken over primes. Then, using this Dirichlet series with Perron's formula, one obtains:

where "C" is a closed curve encircling all of the roots of <math>\zeta(s).</math>

Conversely, one has the Mellin transform

<math>\frac{1}{\zeta(s)} = s\int_1^\infty \frac{M(x)}{x^{s+1}}\,dx</math>

which holds for <math>Re(s)>1</math>.

A good evaluation, at least asymptotically, would be to obtain, by the method of steepest descent, an inequality:

assuming that there are not multiple non-trivial roots of <math> \zeta (\rho) </math> you have the "exact formula" by residue theorem:

<math> \frac{1}{2 \pi i} \oint _ {C}ds \frac{x^s}{s \zeta (s)} = \sum _ {\rho} \frac{x^{\rho}}{\rho \zeta '(\rho)}-2+\sum_{n=1}^{\infty} \frac{ (-1)^{n-1} (2\pi )^{2n}}{(2n)! n \zeta(2n+1)x^{2n}} </math>

[edit] Calculation

The Mertens function has been computed for an increasing range of n.

Person	Year	Limit
Mertens	1897	10⁴
von Sterneck	1897	1.5 x 10⁵
von Sterneck	1901	5 x 10⁵
von Sterneck	1912	5 x 10⁶
Neubauer	1963	10⁸
Cohen and Dress	1979	7.8 x 10⁹
Dress	1993	10¹²
Lioen and van der Lune	1994	10¹³
Kotnik and van der Lune	2003	10¹⁴

[edit] References

F. Mertens, "Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, IIa 106, (1897) 761-830.
A. M. Odlyzko and H.J.J. te Riele, "Disproof of the Mertens Conjecture", Journal für die reine und angewandte Mathematik 357, (1985) pp. 138-160.
Weisstein, Eric W., Mertens function at MathWorld.
Values of the Mertens function for the first 50 n are given by SIDN A002321
Values of the Mertens function for the first 2500 n are given by PrimeFan's Mertens Values Page ca:Funció de Mertens