Mertens function

Mertens function to n=10 thousand

Mertens function to n=10 million

In number theory, the Mertens function is

$M(n)=\sum _{1\leq k\leq n}\mu (k)$

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 $M(x)=o(x^{{\frac {1}{2}}+\epsilon })$. 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.

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 $\zeta (s)$ 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 $\zeta (s).$

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 $\zeta (\rho )$ you have the "exact formula" by residue theorem:

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

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

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". 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