Mertens conjecture - Citizendia

In mathematics, the Mertens conjecture is a statement about the behaviour of a certain function as its argument increases. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Conjectured to be true by Mertens in 1897, it was disproved in 1985. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of Franz Mertens ( March 20, 1840 - March 5, 1927) was a German Mathematician. The Mertens conjecture was interesting, because if true, it would have proved that the famous Riemann hypothesis were also true. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved However, Merten's conjecture being disproved did not, conversely, mean that the Riemann hypothesis was also untrue.

1 Definition
2 Disposition of the conjecture
3 Connection to the Riemann hypothesis
4 References

Definition

In number theory, if we define the Mertens function as

$M(n) = \sum_{1\le k \le n} \mu(k)$

where μ(k) is the Möbius function, then the Mertens conjecture is that

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

Disposition of the conjecture

Stieltjes claimed in 1885 to have proven a weaker result, namely that ${M(n)\over \sqrt{n}}$ was bounded, but did not publish a proof. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Number theory, the Mertens function is M(n = \sum_{1\le k \le n} \mu(k where μ(k is the Möbius function. For the rational functions defined on the complex numbers see Möbius transformation. This article is about Thomas Joannes Stieltjes (pronounced 'stiltʃəs the mathematician In Mathematics, a function f defined on some set X with real or complex values is called bounded, if the set He may have found the reasoning supporting his result was flawed.

In 1985, te Riele and Odlyzko proved the Mertens conjecture false. Hermanus Johannes Joseph te Riele (born January 5, 1947) is a mathematician at CWI in Amsterdam with a specialization in algorithms in discrete Andrew Odlyzko is a mathematician who is the head of the University of Minnesota 's Digital Technology Center It was later shown that there is a counterexample between 10¹⁴ and exp(3. In Logic, and especially in its applications to Mathematics and Philosophy, a counterexample is an exception to a proposed general rule i 21×10⁶⁴), with the upper bound having been lowered to exp(1. 59×10⁴⁰) since, but no counterexample is explicitly known. The boundedness claim made by Stieltjes, while remarked upon as "very unlikely" in the 1985 paper, has not been disproven (as of 2005).

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,

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

valid in the region $\Re(s) > 1$ . In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in We can rewrite this as a Stieltjes integral

$\frac{1}{\zeta(s)} = \int_0^{\infty} x^{-s}\,dM$

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

$\frac{1}{s \zeta(s)} = \left\{ \mathcal{M} M \right\}(-s) = \int_0^\infty x^{-s} M(x)\, \frac{dx}{x}.$

Using the Mellin inversion theorem we now can express M in terms of 1/ζ as

$M(x) = \frac{1}{2 \pi i} \int_{\sigma-is}^{\sigma+is} \frac{x^s}{s \zeta(s)}\, ds$

which is valid for 1 < σ < 2, and valid for 1/2 < σ < 2 on the Riemann hypothesis. In Mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes In Mathematics, the Mellin transform is an Integral transform that may be regarded as the multiplicative version of the Two-sided Laplace transform In Mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions underwhich the inverse Mellin transform, or equivalently the From this, the Mellin transform integral must be convergent, and hence M(x) must be o(x^e) for every exponent greater than 1/2, but not little-o when e equals 1/2. From this it follows that " $M(x) \ne o(x^\frac12)$ but $M(x) = o(x^{\frac12+\epsilon})$ " is equivalent to the Riemann hypothesis, would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that $M(x) = O(x^\frac12)$ .

References

F. Mertens, "Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, IIa 106, (1897) 761-830.
A. M. Odlyzko and H.J.J. te Riele, "Disproof of the Mertens Conjecture", Journal für die reine und angewandte Mathematik 357, (1985) pp. Andrew Odlyzko is a mathematician who is the head of the University of Minnesota 's Digital Technology Center Hermanus Johannes Joseph te Riele (born January 5, 1947) is a mathematician at CWI in Amsterdam with a specialization in algorithms in discrete 138-160.
T. Kotnik and H. J. J. te Riele, "The Mertens Conjecture Revisited", Proceedings of the 7th Algorithmic Number Theory Symposium (2006), LNCS 4076, pp. 156-167.
Eric W. Weisstein, Mertens conjecture at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

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Contents

Definition

Disposition of the conjecture

Connection to the Riemann hypothesis

References