Mertens conjecture

In number theory, if we define the Mertens function as

$M(n)=\sum _{1\leq k\leq n}\mu (k)$

where μ(k) is the Möbius function, then the Mertens conjecture is that

$\left|M(n)\right|<{\sqrt {n}}$

Stieltjes claimed in 1885 to have proved that $M(n)$ always stayed between two fixed bounds, but did not publish a proof, probably because he found out his proof was flawed.

The Mertens conjecture is interesting, because if it was true, it would mean that the famous Riemann hypothesis is also true. However, in 1985, te Riele and Odlyzko proved the Mertens conjecture false. However, the claim made by Stieltjes, while remarked upon as "very unlikely" in the 1985 paper, has not been disproven (as of 2004).

The connection to the Riemann hypothesis is based on the fact that we can derive the result

${\frac {1}{\zeta (z)}}=z\int _{1}^{\infty }{\frac {M(x)}{x^{z+1}}}dx$

where ζ(z) is the Riemann zeta function. The Mertens conjecture would mean that this integral converges for Re(z) > 1/2, which in turn would imply that 1/ζ(z) is defined for Re(z) > 1/2 and by symmetry for Re(z) < 1/2. Thus the only zeros of ζ(z) would be at Re(z) = 1/2, which is the Riemann hypothesis.

External links

• Mertens conjecture at MathWorld