Mertens function

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Mertens function to n=10 thousand
Mertens function to n=10 thousand
Mertens function to n=10 million
Mertens function to n=10 million

In number theory, the Mertens function is

M(n) = \sum_{1\le k \le n} \mu(k)

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 M(x) = o(x^{\frac12 + \epsilon}). 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, o refers to little-o notation.

[edit] Integral representations

Using the Euler product one finds that

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

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

\frac{1}{2\pi i}\oint_{C}ds \frac{x^{s}}{s\zeta(s) }=M(x)

where "C" is a closed curve encircling all of the roots of ζ(s).

Conversely, one has the Mellin transform

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

which holds for Re(s) > 1.

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

\oint_{C}dsF(s)e^{st} \sim M(e^{t})

assuming that there are not multiple non-trivial roots of ζ(ρ) you have the "exact formula" by residue theorem:

\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}}

[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.
  • Eric W. Weisstein, 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
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