Mertens conjecture

The graph shows the Mertens function <math>M(n)</math> and the square roots <math>\pm \sqrt{n}</math> for <math>n \le 10,000</math>. After computing these values, Mertens conjectured that the absolute value of <math>M(n)</math> is always bounded by <math>\sqrt{n}</math>. This hypothesis, known as the Mertens conjecture, was disproved in 1985 by Andrew Odlyzko and Herman te Riele.

In mathematics, the Mertens conjecture is the statement that the Mertens function <math>M(n)</math> is bounded by <math>\pm\sqrt{n}</math>. 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 Template:Harvs), and again in print by Template:Harvs, and disproved by Template:Harvs. It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor.

DefinitionEdit

In number theory, the Mertens function is defined as

where μ(k) is the Möbius function; the Mertens conjecture is that for all n > 1,

Disproof of the conjectureEdit

Stieltjes claimed in 1885 to have proven a weaker result, namely that <math>m(n) := M(n)/\sqrt{n}</math> was bounded, but did not publish a proof.<ref>Template:Cite book</ref> (In terms of <math>m(n)</math>, the Mertens conjecture is that <math> -1 < m(n) < 1 </math>.)

In 1985, Andrew Odlyzko and Herman te Riele proved the Mertens conjecture false using the Lenstra–Lenstra–Lovász lattice basis reduction algorithm:<ref name= disproof>Template:Citation</ref><ref name=HBI1889>Sandor et al (2006) pp. 188–189.</ref>

<math>\liminf m(n) < -1.009 </math> Template:Pad and Template:Pad <math> \limsup m(n) > 1.06.</math>

It was later shown that the first counterexample appears below <math>e^{3.21\times10^{64}} \approx 10^{1.39\times10^{64}}</math><ref>Template:Cite journal </ref> but above 10¹⁶.<ref name=H16>Template:Cite arXiv</ref> The upper bound has since been lowered to <math>e^{1.59\times10^{40}}</math><ref>Kotnik and Te Riele (2006).</ref> or approximately <math>10^{6.91\times10^{39}},</math> and then again to <math>e^{1.017\times10^{29}} \approx 10^{4.416\times10^{28}}</math>.<ref>Template:Cite arXiv</ref> In 2024, Seungki Kim and Phong Nguyen lowered the bound to <math>e^{1.96\times10^{19}} \approx 10^{8.512\times10^{18}}</math>,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> but no explicit counterexample is known.

The law of the iterated logarithm states that if Template:Mvar is replaced by a random sequence of +1s and −1s then the order of growth of the partial sum of the first Template:Mvar terms is (with probability 1) about Template:Nowrap which suggests that the order of growth of Template:Math might be somewhere around Template:Math. The actual order of growth may be somewhat smaller; in the early 1990s Steve Gonek conjectured<ref name=NGonMertens /> that the order of growth of Template:Math was <math>(\log\log\log n)^{5/4},</math> which was affirmed by Ng (2004), based on a heuristic argument, that assumed the Riemann hypothesis and certain conjectures about the averaged behavior of zeros of the Riemann zeta function.<ref name=NGonMertens>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In 1979, Cohen and Dress<ref>Cohen, H. and Dress, F. 1979. “Calcul numérique de Mx)” 11–13. [Cohen et Dress 1979], Rapport, de I'ATP A12311 ≪ Informatique 1975 ≫</ref> found the largest known value of <math>m(n) \approx 0.570591</math> for Template:Math and in 2011, Kuznetsov found the largest known negative value (largest in the sense of absolute value) <math>m(n) \approx -0.585768</math> for Template:Math<ref>Template:Cite arXiv</ref> In 2016, Hurst computed Template:Math for every Template:Math but did not find larger values of Template:Math.<ref name=H16></ref>

In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of Template:Mvar for which Template:Nowrap but without giving any specific value for such an Template:Mvar.<ref>Kotnik & te Riele (2006).</ref> In 2016, Hurst made further improvements by showing

<math>\liminf m(n) < -1.837625 </math> Template:Pad and Template:Pad <math> \limsup m(n) > 1.826054.</math>

Connection to the Riemann hypothesisEdit

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

<math>\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s},</math>

valid in the region <math>\mathcal{Re}(s) > 1</math>. We can rewrite this as a Stieltjes integral

<math>\frac{1}{\zeta(s)} = \int_0^\infty x^{-s} dM(x)</math>

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

<math>\frac{1}{s \zeta(s)} = \left\{ \mathcal{M} M \right\}(-s)

= \int_0^\infty x^{-s} M(x)\, \frac{dx}{x}.</math>

Using the Mellin inversion theorem we now can express Template:Mvar in terms of Template:Frac as

<math>M(x) = \frac{1}{2 \pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{x^s}{s \zeta(s)}\,ds</math>

which is valid for Template:Math, and valid for Template:Math on the Riemann hypothesis. From this, the Mellin transform integral must be convergent, and hence Template:Math must be Template:Math for every exponent e greater than Template:Sfrac. From this it follows that

<math>M(x) = O\Big(x^{\tfrac{1}{2} + \epsilon}\Big)</math>

for all positive Template:Mvar is equivalent to the Riemann hypothesis, which therefore would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that

<math>M(x) = O\Big(x^\tfrac{1}{2}\Big).</math>

ReferencesEdit

Template:Reflist

External linksEdit

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