Franz Mertens conjectured that $\left| M(n) \right| < \sqrt {n}$ where the Mertens function is defined as $$M(n) = \sum_{i = 1}^n \mu(i),$$ and $\mu$ is the Möbius function.

However, Herman J. J. te Riele and Andrew Odlyzko have proven that there exist counterexamples beyond $10^{13}$ , but have yet to find one specific counterexample.

The Mertens conjecture is related to the Riemann hypothesis, since $$M(x) = O(x^\frac12)$$ is another way of stating the Riemann hypothesis.

Given the Dirichlet series of the reciprocal of the Riemann zeta function, we find that $$\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}$$ is true for $\Re(s) > 1$ . Rewriting as Stieltjes integral, $$\frac{1}{\zeta(s)} = \int_0^{\infty} x^{-s}dM$$ suggests this Mellin transform: $$\frac{1}{s \zeta(s)} = \left\{ \mathcal{M} M \right\}(-s) = \int_0^\infty x^{-s} M(x) \frac{dx}{x}.$$ Then it follows that $$M(x) = \frac{1}{2 \pi i} \int_{\sigma-is}^{\sigma+is} \frac{x^s}{s \zeta(s)} ds$$ for $\frac{1}{2} < \sigma < 2$ .

Bibliography

1

G. H. Hardy and S. Ramanujan, Twelve Lectures on Subjects Suggested by His Life and Work 3rd ed. New York: Chelsea, p. 64 (1999)

2

A. M. Odlyzko and H. J. J. te Riele, ``Disproof of the Mertens Conjecture." J. reine angew. Math. 357, pp. 138 - 160 (1985)