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[[File:Mertens.svg|thumb|right|Mertens function to n=10,000]]
[[File:Mertens-big.svg|thumb|right|Mertens function to n=10,000,000]]
In [[number theory]], the '''Mertens function''' is defined for all  positive [[integer]]s ''n'' as


:<math>M(n) = \sum_{k=1}^n \mu(k)</math>
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where μ(k) is the [[Möbius function]]. The function is named in honour of [[Franz Mertens]].
 
Less formally, ''M''(''n'') is the count of [[square-free integer]]s up to ''n'' that have an even number of prime factors, minus the count of those that have an odd number. The
Mertens function slowly grows in positive and negative directions both on average and in peak value, oscillating in an apparently chaotic manner passing through zero when n has the values
:2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, ... {{OEIS|id=A028442}}.
 
Because the Möbius function only takes the values &minus;1, 0, and +1, the Mertens function moves slowly and  there is no ''n'' such that |''M''(''n'')| >&nbsp;''n''. The [[Mertens conjecture]] went further, stating that there would be no ''n'' where the absolute value of the Mertens function exceeds the square root of ''n''. The Mertens conjecture was proven false in 1985 by [[Andrew Odlyzko]] and [[Herman te Riele]]. However, the [[Riemann hypothesis]] is equivalent to a weaker conjecture on the growth of ''M''(''n''), namely ''M''(''n'') = ''O''(''n''<sup>1/2 + ε</sup>). Since high values for ''M''(''n'') grow at least as fast as the square root of ''n'', this puts a rather tight bound on its rate of growth. Here, ''O'' refers to [[Big O notation]].
 
The above definition can be extended to [[real numbers]] as follows:
 
:<math>M(x) = \sum_{1\le k \le x} \mu(k).</math>
 
==Representations==
 
===As an integral===
{{Cleanup|section|date=December 2009}}
Using the [[Euler product]] one finds that
 
:<math> \frac{1}{\zeta(s) }= \prod_{p} (1-p^{-s})= \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}</math>
 
where <math>\zeta(s)</math> is the [[Riemann zeta function]] and the product is taken over primes. Then, using this [[Dirichlet series]] with [[Perron's formula]], one obtains:
 
:<math> \frac{1}{2\pi i}\oint_{C} \frac{x^{s}}{s\zeta(s)} \, ds = M(x) </math>
 
where ''C'' is a [[Contour integration|closed curve]] encircling all of the roots of <math>\zeta(s).</math>
 
Conversely, one has the [[Mellin transform]]
 
:<math>\frac{1}{\zeta(s)} = s\int_1^\infty \frac{M(x)}{x^{s+1}}\,dx</math>
 
which holds for <math> \mathrm{Re}(s)>1</math>.
 
A curious relation given by Mertens himself involving the second [[Chebyshev function]] is
 
: <math> \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) + \cdots. </math>
 
A good evaluation, at least asymptotically, would be to obtain, by the [[method of steepest descent]], the inequality
 
:<math> \oint_C F(s)e^{st} \, ds \sim M(e^{t}). </math>
 
Assuming that there are not multiple non-trivial roots of <math> \zeta (\rho) </math> we have the "exact formula" by the [[residue theorem]]:
 
: <math>  \frac{1}{2 \pi i} \oint_C \frac{x^s}{s \zeta (s)} \, ds = \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}}. </math>
 
[[Hermann Weyl|Weyl]] conjectured that the Mertens function satisfied the approximate functional-differential equation
 
: <math> \frac{y(x)}{2}-\sum_{r=1}^N \frac{B_{2r}}{(2r)!}D_t^{2r-1} y \left(\frac{x}{t+1}\right) + x\int_0^x \frac{y(u)}{u^{2}} \, du = x^{-1}H(\log x) </math>
 
where ''H''(''x'') is the [[Heaviside step function]], ''B'' are [[Bernoulli number]]s and all derivatives with respect to ''t'' are evaluated at ''t'' = 0.
 
[[Edward Charles Titchmarsh|Titchmarsh]](1960), and later J. Garcia  provided a Trace formula involving a sum over the Möbius function and zeros of Riemann Zeta in the form
 
: <math> \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{2n}}{(2n)! \zeta(2n+1)}\int_{-\infty}^{\infty}g(x) e^{-x(2n+1/2)} \, dx, </math>
 
where 't' sums over the imaginary parts of nontrivial zeros, and (g, h) are related by a Fourier transform, such that
:<math> 2 \pi g(x)= \int_{-\infty}^{\infty}h(u)e^{iux} \, du. </math>
 
===As a sum over Farey sequences===
Another formula for the Mertens function is
 
:<math>M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a}</math> &nbsp; where &nbsp; <math> \mathcal{F}_n</math> &nbsp; is the [[Farey sequence]] of order ''n''.
 
This formula is used in the proof of the [[Farey sequence#Riemann hypothesis|Franel–Landau theorem]].<ref>Edwards, Ch. 12.2</ref>
 
===As a determinant===
''M''(''n'') is the [[determinant]] of the ''n''&nbsp;&times;&nbsp;''n'' [[Redheffer matrix]], a [[(0,1) matrix]] in which
''a''<sub>''ij''</sub> is 1 if either ''j'' is 1 or ''i'' divides ''j''.
 
== Calculation ==
 
Neither of the methods mentioned previously leads to practical algorithms to calculate the Mertens function.
Using sieve methods similar to those used in prime counting, the Mertens function has been computed for an increasing range of ''n''.
 
{| cellpadding="5" border="1"
|- style="background:gray; color:white"
| Person || Year || Limit
|- style="vertical-align:bottom;"
| Mertens || 1897 || 10<sup>4</sup>
|- style="vertical-align:bottom;"
| von Sterneck || 1897 || 1.5{{e|5}}
|- style="vertical-align:bottom;"
| von Sterneck || 1901 || 5{{e|5}}
|- style="vertical-align:bottom;"
| von Sterneck || 1912 || 5{{e|6}}
|- style="vertical-align:bottom;"
| Neubauer || 1963 || 10<sup>8</sup>
|- style="vertical-align:bottom;"
| Cohen and Dress || 1979 || 7.8{{e|9}}
|- style="vertical-align:bottom;"
| Dress || 1993 || 10<sup>12</sup>
|- style="vertical-align:bottom;"
| Lioen and van de Lune || 1994 || 10<sup>13</sup>
|- style="vertical-align:bottom;"
| Kotnik and van de Lune || 2003 || 10<sup>14</sup>
|}
 
The Mertens function for all integer values up to ''N'' may be computed in ''O(N<sup>2/3+ε</sup>)'' time, while better methods are known. Elementary algorithms exist to compute isolated values of ''M(N)'' in ''O(N<sup>2/3</sup>*(ln ln(N))<sup>1/3</sup>)'' time.
 
See {{OEIS2C|A084237}} for values of M(N) at powers of 10.
 
==Notes==
{{Reflist}}
 
==See also==
* [[Perron's formula]]
 
== References ==
*{{cite book
  | last = Edwards
  | first = Harold | authorlink = Harold Edwards (mathematician)
  | title = Riemann's Zeta Function
  | publisher = Dover
  | location = Mineola, New York
  | year = 1974
  | isbn = 0-486-41740-9}}
*F. Mertens, "Über eine zahlentheoretische Funktion", ''Akademie Wissenschaftlicher Wien Mathematik-Naturlich'' Kleine Sitzungsber, IIa '''106''', (1897) 761–830.
*[[A. M. Odlyzko]] and [[Herman te Riele]], "[http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf Disproof of the Mertens Conjecture]", ''Journal für die reine und angewandte Mathematik'' '''357''', (1985) pp.&nbsp;138–160.
*{{mathworld|urlname=MertensFunction|title=Mertens function}}
* Jose Javier Garcia Moreta "http://www.prespacetime.com/index.php/pst/issue/view/42 Borel Resummation & the Solution of Integral Equations
*{{SloanesRef |sequencenumber=A002321|name=Mertens's function}}
*Deléglise, M. and Rivat, J. "Computing the Summation of the Möbius Function." Experiment. Math. 5, 291-295, 1996. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.em/1047565447
 
{{DEFAULTSORT:Mertens Function}}
[[Category:Arithmetic functions]]

Latest revision as of 07:03, 27 October 2014


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