In number theory, the Mertens function is

where μ(k) is the Möbius function.

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 . 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.

External links

• 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