Mertens function

In number theory, the Mertens function is

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

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.

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

Categories: Number theory