Mertens conjecture

In mathematics, the Mertens conjecture is the statement that the Mertens function ${\displaystyle M(n)}$ is bounded by ${\displaystyle \pm {\sqrt {n}}}$. Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in Stieltjes (1905)), and again in print by Franz Mertens (1897), and disproved by Andrew Odlyzko and Herman te Riele (1985). It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor.