Mertens conjecture

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The Mertens conjecture is a statement about the behaviour of a certain function as its argument increases. Conjectured to be true by Mertens in 1897, it was disproved in 1985. The Mertens conjecture was interesting, because if true, it would have meant that the famous Riemann hypothesis was also true. However, Merten's conjecture being disproved did not, conversely, mean that the Riemann hypothesis was also untrue.

1 Definition
2 Disposition of the conjecture
3 Connection to the Riemann hypothesis
4 References

[edit] Definition

In number theory, if we define the Mertens function as

$M(n) = \sum_{1\le k \le n} \mu(k)$

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

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

[edit] Disposition of the conjecture

Stieltjes claimed in 1885 to have proved a weaker result, namely that ${M(n)\over \sqrt{n}}$ always stayed between two fixed bounds, but did not publish a proof, possibly because he found out his proof was flawed.

In 1985, te Riele and Odlyzko proved the Mertens conjecture false. It was later shown that there is a counterexample between 10¹⁴ and 3.21×10⁶⁴, with the upper bound having been lowered to 1.59×10⁴⁰ since, but no counterexample is explicitly known. The boundedness claim made by Stieltjes, while remarked upon as "very unlikely" in the 1985 paper, has not been disproven (as of 2005).

[edit] Connection to the Riemann hypothesis

The connection to the Riemann hypothesis is based on the Dirichlet series for the reciprocal of the Riemann zeta function,

$\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s},$

valid in the region $\Re(s) > 1$ . We can rewrite this as a Stieltjes integral

$\frac{1}{\zeta(s)} = \int_0^{\infty} x^{-s}\,dM$

and after integrating by parts, obtain the reciprocal of the zeta function as a Mellin transform

$\frac{1}{s \zeta(s)} = \left\{ \mathcal{M} M \right\}(-s) = \int_0^\infty x^{-s} M(x)\, \frac{dx}{x}.$

Using the Mellin inversion theorem we now can express M in terms of 1/ζ as

$M(x) = \frac{1}{2 \pi i} \int_{\sigma-is}^{\sigma+is} \frac{x^s}{s \zeta(s)}\, ds$

which is valid for 1 < σ < 2, and valid for 1/2 < σ < 2 on the Riemann hypothesis. From this, the Mellin transform integral must be convergent, and hence M(x) must be o(x^e) for every exponent greater than 1/2, but not little-o when e equals 1/2. From this it follows that " $M(x) \ne o(x^\frac12)$ but $M(x) = o(x^{\frac12+\epsilon})$ " is equivalent to the Riemann hypothesis, would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that $M(x) = O(x^\frac12)$ .

[edit] References

F. Mertens, "Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, IIa 106, (1897) 761-830.
A. M. Odlyzko and H.J.J. te Riele, "Disproof of the Mertens Conjecture", Journal für die reine und angewandte Mathematik 357, (1985) pp. 138-160.
T. Kotnik and H.J.J. te Riele, "The Mertens Conjecture Revisited", Proceedings of the 7th Algorithmic Number Theory Symposium (2006), LNCS 4076, pp. 156-167.
Weisstein, Eric W., Mertens conjecture at MathWorld.