In
mathematics, the Mertens conjecture is the statement that the
Mertens function is bounded by
. 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
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermite p ...
(reprinted in ), 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(k) is the
Möbius function; the Mertens conjecture is that for all ''n'' > 1,
:
Disproof of the conjecture
Stieltjes claimed in 1885 to have proven a weaker result, namely that
was
bounded, but did not publish a proof. (In terms of
, the Mertens conjecture is that
.)
In 1985,
Andrew Odlyzko
Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish- American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career in ...
and
Herman te Riele
Hermanus Johannes Joseph te Riele (born 5 January 1947) is a Dutch mathematician at CWI in Amsterdam with a specialization in computational number theory. He is known for proving the correctness of the Riemann hypothesis for the first 1.5 billi ...
proved the Mertens conjecture false using the
Lenstra–Lenstra–Lovász lattice basis reduction algorithm:
[Sandor et al (2006) pp. 188–189.]
:
and
It was later shown that the first
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
appears below
but above 10
16.
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 (with probability 1) 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 Steve Gonek conjectured
that the order of growth of was
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
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 .
[Kotnik & te Riele (2006).] In 2016, Hurst made further improvements by showing
:
and
Connection to the Riemann hypothesis
The connection to the Riemann hypothesis is based on the
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in ana ...
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
:
References
Further reading
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External links
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Analytic number theory
Disproved conjectures