Mertens conjecture

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In mathematics, 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 proved that the famous Riemann hypothesis were also true. However, Merten's conjecture being disproved did not, conversely, mean that the Riemann hypothesis was also untrue.

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[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 proven a weaker result, namely that {M(n)\over \sqrt{n}} was bounded, but did not publish a proof. He may have found the reasoning supporting his result 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 exp(3.21×1064), with the upper bound having been lowered to exp(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(xe) 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