Mertens conjecture

The graph shows the Mertens function $M(n)$ and the square roots $\pm \sqrt{n}$ for $n\le 10,000$. After computing these values, Mertens conjectured that the absolute value of $M(n)$ is always bounded by $\sqrt{n}$. 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 $M(n)$ is bounded by $\pm \sqrt{n}$. 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.

Definition

In number theory, the Mertens function is defined as

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

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

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

Disproof of the conjecture

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

In 1985, Andrew Odlyzko and Herman te Riele proved the Mertens conjecture false using the Lenstra–Lenstra–Lovász lattice basis reduction algorithm:

$\mathrm{lim\; inf}m(n)<-1.009$ Template:Pad and Template:Pad $\mathrm{lim\; sup}m(n)>1.06.$

It was later shown that the first counterexample appears below $e^{3.21\times 10^{64}}\approx 10^{1.39\times 10^{64}}$ but above 10¹⁶. The upper bound has since been lowered to $e^{1.59\times 10^{40}}$ or approximately $10^{6.91\times 10^{39}},$ and then again to $e^{1.017\times 10^{29}}\approx 10^{4.416\times 10^{28}}$. In 2024, Seungki Kim and Phong Nguyen lowered the bound to $e^{1.96\times 10^{19}}\approx 10^{8.512\times 10^{18}}$, 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 that the order of growth of Template:Math was $(\log \log \log n{)}^{5/4},$ 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.

In 1979, Cohen and Dress found the largest known value of $m(n)\approx 0.570591$ for Template:Math and in 2011, Kuznetsov found the largest known negative value (largest in the sense of absolute value) $m(n)\approx -0.585768$ for Template:Math In 2016, Hurst computed Template:Math for every Template:Math but did not find larger values of Template:Math.

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. In 2016, Hurst made further improvements by showing

$\mathrm{lim\; inf}m(n)<-1.837625$ Template:Pad and Template:Pad $\mathrm{lim\; sup}m(n)>1.826054.$

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,

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

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

$\frac{1}{\zeta (s)}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{-s}dM(x)$

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

$\frac{1}{s\zeta (s)}=\left\{ℳM\right\}(-s)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{-s}M(x)\frac{dx}{x}.$

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

$M(x)=\frac{1}{2\pi i}\underset{\sigma -i\mathrm{\infty }}{\overset{\sigma +i\mathrm{\infty }}{\int }}\frac{x^s}{s\zeta (s)}ds$

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

$M(x)=O\left(x^{\frac{1}{2}+ϵ}\right)$

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

$M(x)=O\left(x^{\frac{1}{2}}\right).$

Further reading

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External links

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