Mertens conjecture


In mathematics, the Mertens conjecture is the statement that the Mertens function is bounded by. Although now disproven, it has been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite, and again in print by, and disproved by.
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, we define the Mertens function as
where μ is the Möbius function; the Mertens conjecture is that for all n > 1,

Disproof of the conjecture

claimed in 1885 to have proven a weaker result, namely that was bounded, but did not publish a proof.
In 1985, Andrew Odlyzko and Herman te Riele proved the Mertens conjecture false using the Lenstra–Lenstra–Lovász lattice basis reduction algorithm:
It was later shown that the first counterexample appears below but above 1014. The upper bound has since been lowered to or approximately but no explicit counterexample is known.
The law of the iterated logarithm states that if is replaced by a random sequence of +1s and −1s then the order of growth of the partial sum of the first terms is about which suggests that the order of growth of might be somewhere around. The actual order of growth may be somewhat smaller; in the early 1990s Gonek conjectured that the order of growth of was, which was affirmed by Ng, 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 for and in 2011, Kuznetsov found the largest known negative value for In 2016, Hurst computed for every but did not find larger values of.
In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of for which but without giving any specific value for such an. In 2016, Hurst made further improvements by showing

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,
valid in the region. We can rewrite this as a
Stieltjes integral
and after integrating by parts, obtain the reciprocal of the zeta function
as a Mellin transform
Using the Mellin inversion theorem we now can express in terms of as
which is valid for, and valid for on the Riemann hypothesis.
From this, the Mellin transform integral must be convergent, and hence
must be for every exponent e greater than. From this it follows that
for all positive is equivalent to the Riemann hypothesis, which therefore would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that