Mertens conjecture

In mathematics, the Mertens conjecture is the false statement that the Mertens function M(n) is bounded by √n, which implies the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes in an 1885 letter to Charles Hermite (reprinted in Stieltjes 1905) and Mertens (1897), and disproved by Odlyzko & te Riele (1985). It is a striking example of a mathematical proof contradicting a large amount of computational evidence in favor of a conjecture.

In number theory, we define the Mertens function as

where μ(k) is the Möbius function; the Mertens conjecture is that for all n > 1,

Stieltjes claimed in 1885 to have proven a weaker result, namely that ${\displaystyle m(n)=M(n)/{\sqrt {n}}}$ was bounded, but did not publish a proof. (In terms of ${\displaystyle m(n)}$, the Mertens conjecture is that ${\displaystyle -1<m(n)<1}$.)

...
Wikipedia