Mertens conjecture

In number theory, if we define the Mertens function as

[itex]M(n) = \sum_{1\le k \le n} \mu(k)[itex]

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

[itex]\left| M(n) \right| < \sqrt { n }[itex]

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

The Mertens conjecture was interesting, because if true, it would have meant that the famous Riemann hypothesis was also true. However, in 1985, te Riele and Odlyzko proved the Mertens conjecture false. The boundedness claim made by Stieltjes, while remarked upon as "very unlikely" in the 1985 paper, has not been disproven (as of 2005).

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

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

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

[itex]\frac{1}{\zeta(s)} = \int_0^{\infty} x^{-s}dM[itex]

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

[itex]\frac{1}{s \zeta(s)} = \left\{ \mathcal{M} M \right\}(-s)

= \int_0^\infty x^{-s} M(x) \frac{dx}{x}[itex]

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

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

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 "[itex]M(x) = \Omega(x^\frac12)[itex] but [itex]M(x) = o(x^{\frac12+\epsilon})[itex]" is equivalent to the Riemann hypothesis, would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that [itex]M(x) = O(x^\frac12)[itex].

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