Mertens function - Citizendia

Mertens function to n=10 thousand

Mertens function to n=10 million

In number theory, the Mertens function is

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

where μ(k) is the Möbius function. 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 For the rational functions defined on the complex numbers see Möbius transformation. The function is named in honour of Franz Mertens. Franz Mertens ( March 20, 1840 - March 5, 1927) was a German Mathematician.

Less formally, M(n) is the count of square-free integers up to n that have an even number of prime factors, minus the count of those that have an odd number. In Mathematics, a square-free, or quadratfrei, Integer is one divisible by no perfect square, except 1 M(n) = 0 for the n values

2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, . . . (sequence A028442 in OEIS)

Because the Möbius function has only the return values -1, 0 and +1, it's obvious that the Mertens function moves slowly and that there is no k such that |M(k)| > k. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences The Mertens conjecture went even further, stating that there would be no k where the absolute value of the Mertens function exceeds the square root of k. In Mathematics, the Mertens conjecture is a statement about the behaviour of a certain function as its argument increases The Mertens conjecture was disproven in 1985. However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(k), namely $M(k) = o(k^{\frac12 + \epsilon})$ . 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 Since high values for M grow at least as fast as the square root of k, this puts a rather tight bound on its rate of growth. Here, $o$ refers to little-o notation. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments

Integral representations

Using the Euler product one finds that

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

where $ζ(s)$ is the Riemann zeta function and the product is taken over primes. In Number theory, an Euler product is an Infinite product expansion indexed by Prime numbers p, of a Dirichlet series. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in Then, using this Dirichlet series with Perron's formula, one obtains:

$\frac{1}{2\pi i}\oint_{C}ds \frac{x^{s}}{s\zeta(s) }=M(x)$

where "C" is a closed curve encircling all of the roots of $ζ(s). In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and In Mathematics, and more particularly in Analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetical$

Conversely, one has the Mellin transform

$\frac{1}{\zeta(s)} = s\int_1^\infty \frac{M(x)}{x^{s+1}}\,dx$

which holds for $R e (s) > 1$ . In Mathematics, the Mellin transform is an Integral transform that may be regarded as the multiplicative version of the Two-sided Laplace transform

A curious relation given by Mertens himself involving Chebyshev function is:

$\Psi (x) = -M\left(\frac{x}{2}\right)\log(2)-M\left(\frac{x}{3}\right)\log(3)-M\left(\frac{x}{4}\right)\log(4)+......$

A good evaluation, at least asymptotically, would be to obtain, by the method of steepest descent, an inequality:

$\oint_{C}dsF(s)e^{st} \sim M(e^{t})$

assuming that there are not multiple non-trivial roots of $ζ(ρ)$ you have the "exact formula" by residue theorem:

$\frac{1}{2 \pi i} \oint _ {C}ds \frac{x^s}{s \zeta (s)} = \sum _ {\rho} \frac{x^{\rho}}{\rho \zeta '(\rho)}-2+\sum_{n=1}^{\infty} \frac{ (-1)^{n-1} (2\pi )^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$

Weyl conjectured that Mertens function satisfied the approximate functional-differential equation

$(1/2)y(x)-\sum_{r=1}^{N} \frac{B_{2r}}{(2r)!}D_{t}^{2r-1}y(\frac{x}{t+1})+x\int_{0}^{x}du \frac{y(u)}{u^{2}}=x^{-1}H(logx)$

where H(x) is the Heaviside step function, B are Bernoulli numbers and all derivatives with respect to t are evaluated at t = 0. For the optimization method called "steepest descent" see Gradient descent. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. The Heaviside step function, H, also called the unit step function, is a discontinuous function whose value is zero for negative In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory.

Calculation

The Mertens function has been computed for an increasing range of n.

Person	Year	Limit
Mertens	1897	10⁴
von Sterneck	1897	1. 5 x 10⁵
von Sterneck	1901	5 x 10⁵
von Sterneck	1912	5 x 10⁶
Neubauer	1963	10⁸
Cohen and Dress	1979	7. 8 x 10⁹
Dress	1993	10¹²
Lioen and van der Lune	1994	10¹³
Kotnik and van der Lune	2003	10¹⁴

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.
Eric W. Weisstein, Mertens function 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
Values of the Mertens function for the first 10,000 n are given by A002321.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |