Mertens conjecture
Encyclopedia
In mathematics
, the Mertens conjecture is the incorrect statement that the Mertens function M(n) is bounded by √n, which implies the Riemann hypothesis
. It was conjectured by Stieltjes in a 1885 letter to Hermite (reprinted in ) and , and disproved by .
, if we define the Mertens function as
where μ(k) is the Möbius function
, then the Mertens conjecture is that for all n > 1,
was bounded
, but did not publish a proof.
In 1985, Andrew Odlyzko
and Herman te Riele proved the Mertens conjecture false. It was later shown that the first counterexample
appears below exp(3.21) (Pintz
1987) but above 1014 (Kotnik and Van de Lune 2004). The upper bound has since been lowered to exp(1.59) (Kotnik and Te Riele 2006), but no counterexample is explicitly known. The boundedness claim made by Stieltjes, while remarked upon as "very unlikely" in the 1985 paper cited above, has not been disproven . The law of the iterated logarithm
states that if
μ is replaced by a random sequence of 1s and −1s then the order of growth of the partial sum of the first n terms is (with probability 1) about (n log log n)1/2, which suggests that the order of growth of M(n)/n1/2 might be somewhere around (log log n)1/2.
In 1979 Cohen and Dress found the largest known value of |M(n)|/n1/2 =~ 0.570591 for M(7766842813) = 50286. In 2003 Kotnik and van de Lune extended the search to n = 1014 but did not find larger values.
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 M in terms of
1/ζ as
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 e greater than
1/2. From this it follows that
for all positive ε is equivalent to the Riemann hypothesis, which therefore would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Mertens conjecture is the incorrect statement that the Mertens function M(n) is bounded by √n, which implies the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
. It was conjectured by Stieltjes in a 1885 letter to Hermite (reprinted in ) and , and disproved by .
Definition
In number theoryNumber theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, if we define the Mertens function as
where μ(k) is the Möbius function
Möbius function
The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
, then 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 boundedBounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
, but did not publish a proof.
In 1985, Andrew Odlyzko
Andrew Odlyzko
Andrew Michael Odlyzko is a mathematician and a former head of the University of Minnesota's Digital Technology Center.In the field of mathematics he has published extensively on analytic number theory, computational number theory, cryptography, algorithms and computational complexity,...
and Herman te Riele proved the Mertens conjecture false. It was later shown that the first counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
appears below exp(3.21) (Pintz
János Pintz
János Pintz is a Hungarian mathematician working in analytic number theory. He is a fellow of the Rényi Mathematical Institute and is also a member of the Hungarian Academy of Sciences.-Mathematical results:...
1987) but above 1014 (Kotnik and Van de Lune 2004). The upper bound has since been lowered to exp(1.59) (Kotnik and Te Riele 2006), but no counterexample is explicitly known. The boundedness claim made by Stieltjes, while remarked upon as "very unlikely" in the 1985 paper cited above, has not been disproven . The law of the iterated logarithm
Law of the iterated logarithm
In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Y. Khinchin . Another statement was given by A.N...
states that if
μ is replaced by a random sequence of 1s and −1s then the order of growth of the partial sum of the first n terms is (with probability 1) about (n log log n)1/2, which suggests that the order of growth of M(n)/n1/2 might be somewhere around (log log n)1/2.
In 1979 Cohen and Dress found the largest known value of |M(n)|/n1/2 =~ 0.570591 for M(7766842813) = 50286. In 2003 Kotnik and van de Lune extended the search to n = 1014 but did not find larger values.
Connection to the Riemann hypothesis
The connection to the Riemann hypothesis is based on the Dirichlet seriesfor the reciprocal of the Riemann zeta function,
valid in the region
. We can rewrite this as aStieltjes integral
and after integrating by parts, obtain the reciprocal of the zeta function
as a Mellin transform
Mellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
Using the Mellin inversion theorem
Mellin inversion theorem
In mathematics, the Mellin inversion formula tells us conditions underwhich the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function....
we now can express M in terms of
1/ζ as
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 e greater than
1/2. From this it follows that
for all positive ε is equivalent to the Riemann hypothesis, which therefore would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that
.