In mathematics, the Mertens conjecture is the statement that the Mertens function M ( n ) {\displaystyle M(n)} is bounded by ± n {\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.
Definition In number theory, the Mertens function is defined as
M ( n ) = ∑ 1 ≤ k ≤ n μ ( k ) , {\displaystyle M(n)=\sum _{1\leq k\leq n}\mu (k),}
where μ(k) is the Möbius function; the Mertens conjecture is that for all n > 1,
| M ( n ) | < n . {\displaystyle |M(n)|<{\sqrt {n}}.}
Disproof of the conjecture Stieltjes claimed in 1885 to have proven a weaker result, namely that m ( n ) := M ( n ) / n {\displaystyle m(n):=M(n)/{\sqrt {n}}} was bounded, but did not publish a proof. (In terms of m ( n ) {\displaystyle m(n)} , the Mertens conjecture is that − 1 < m ( n ) < 1 {\displaystyle -1<m(n)<1} .) In 1985, Andrew Odlyzko and Herman te Riele proved the Mertens conjecture false using the Lenstra–Lenstra–Lovász lattice basis reduction algorithm:
lim inf m ( n ) < − 1.009 {\displaystyle \liminf m(n)<-1.009} and lim sup m ( n ) > 1.06. {\displaystyle \limsup m(n)>1.06.}
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