In number theory, the Mertens function is

$M(n)\; =\; \backslash sum\_\{1\backslash le\; k\; \backslash le\; n\}\; \backslash mu(k)$

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 $M(x)\; =\; o(x^\{\backslash frac12\; +\; \backslash epsilon\})$. 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

$\backslash frac\{1\}\{\backslash zeta(s)\; \}=\; \backslash prod\_\{p\}\; (1-p^\{-s\})=\; \backslash sum\_\{n=1\}^\{\backslash infty\}\backslash mu\; (n)n^\{-s\}$

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

$\backslash frac\{1\}\{2\backslash pi\; i\}\backslash oint\_\{C\}ds\; \backslash frac\{x^\{s\}\}\{s\backslash zeta(s)\; \}=M(x)$

where "C" is a closed curve encircling all of the roots of $\backslash zeta(s).$

Conversely, one has the Mellin transform

$\backslash frac\{1\}\{\backslash zeta(s)\}\; =\; s\backslash int\_1^\backslash infty\; \backslash frac\{M(x)\}\{x^\{s+1\}\}\backslash ,dx$

which holds for $Re(s)>1$.

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

$\backslash oint\_\{C\}dsF(s)e^\{st\}\; \backslash sim\; M(e^\{t\})$

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.
• 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

Arithmetic functions

Funció de Mertens | Función de Mertens | Fonction de Mertens | 메르텐스 함수 | Funzione di Mertens | Mertensfunctie | Função de Mertens | Mertensova funkcija