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{{Short description|Summatory function of the Möbius function}}
{{More citations needed|talk=y|date=December 2009}}
[[File:Mertens.svg|thumb|right|Mertens function to ''n''&nbsp;=&nbsp;{{val|10,000}}]]
[[File:Mertens-big.svg|thumb|right|Mertens function to ''n''&nbsp;=&nbsp;{{val|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>

where <math>\mu(k)</math> is the [[Möbius function]]. The function is named in honour of [[Franz Mertens]]. This definition can be extended to positive [[real numbers]] as follows:

: <math>M(x) = M(\lfloor x \rfloor).</math>

Less formally, <math>M(x)</math> is the count of [[square-free integer]]s up to ''x'' that have an even number of prime factors, minus the count of those that have an odd number.

The first 143 ''M''(''n'') values are {{OEIS|id=A002321}}

{|class="wikitable" style="text-align:right"
!''M''(''n'')
!+0
!+1
!+2
!+3
!+4
!+5
!+6
!+7
!+8
!+9
!+10
!+11
|-
!0+
|
|1
|0
|&minus;1
|&minus;1
|&minus;2
|&minus;1
|&minus;2
|&minus;2
|&minus;2
|&minus;1
|&minus;2
|-
!12+
|&minus;2
|&minus;3
|&minus;2
|&minus;1
|&minus;1
|&minus;2
|&minus;2
|&minus;3
|&minus;3
|&minus;2
|&minus;1
|&minus;2
|-
!24+
|&minus;2
|&minus;2
|&minus;1
|&minus;1
|&minus;1
|&minus;2
|&minus;3
|&minus;4
|&minus;4
|&minus;3
|&minus;2
|&minus;1
|-
!36+
|&minus;1
|&minus;2
|&minus;1
|0
|0
|&minus;1
|&minus;2
|&minus;3
|&minus;3
|&minus;3
|&minus;2
|&minus;3
|-
!48+
|&minus;3
|&minus;3
|&minus;3
|&minus;2
|&minus;2
|&minus;3
|&minus;3
|&minus;2
|&minus;2
|&minus;1
|0
|&minus;1
|-
!60+
|&minus;1
|&minus;2
|&minus;1
|&minus;1
|&minus;1
|0
|&minus;1
|&minus;2
|&minus;2
|&minus;1
|&minus;2
|&minus;3
|-
!72+
|&minus;3
|&minus;4
|&minus;3
|&minus;3
|&minus;3
|&minus;2
|&minus;3
|&minus;4
|&minus;4
|&minus;4
|&minus;3
|&minus;4
|-
!84+
|&minus;4
|&minus;3
|&minus;2
|&minus;1
|&minus;1
|&minus;2
|&minus;2
|&minus;1
|&minus;1
|0
|1
|2
|-
!96+
|2
|1
|1
|1
|1
|0
|&minus;1
|&minus;2
|&minus;2
|&minus;3
|&minus;2
|&minus;3
|-
!108+
|&minus;3
|&minus;4
|&minus;5
|&minus;4
|&minus;4
|&minus;5
|&minus;6
|&minus;5
|&minus;5
|&minus;5
|&minus;4
|&minus;3
|-
!120+
|&minus;3
|&minus;3
|&minus;2
|&minus;1
|&minus;1
|&minus;1
|&minus;1
|&minus;2
|&minus;2
|&minus;1
|&minus;2
|&minus;3
|-
!132+
|&minus;3
|&minus;2
|&minus;1
|&minus;1
|&minus;1
|&minus;2
|&minus;3
|&minus;4
|&minus;4
|&minus;3
|&minus;2
|&minus;1
|}

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, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, ... {{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 ''x'' such that |''M''(''x'')| >&nbsp;''x''. 
H. Davenport<ref>{{cite journal |last1=Davenport |first1=H.|title=On Some Infinite Series Involving Arithmetical Functions (Ii) |journal=The Quarterly Journal of Mathematics |series=Original Series |volume=8 |issue=1 |pages=313–320 |date=November 1937|doi=10.1093/qmath/os-8.1.313 }}</ref> demonstrated that, for any fixed ''h'', 

: <math> \sum_{n=1}^{x}\mu(n)\exp(i2\pi n\theta) = O\left(\frac{x}{\log^hx}\right)</math>

uniformly in <math>\theta</math>. This implies, for <math>\theta=0</math> that

: <math> M(x) = O\left(\frac{x}{\log^hx}\right)\ .</math>

 
The [[Mertens conjecture]] went further, stating that there would be no ''x'' where the absolute value of the Mertens function exceeds the square root of ''x''. 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''(''x''), namely ''M''(''x'') = ''O''(''x''<sup>1/2 + ε</sup>).  Since high values for ''M''(''x'') grow at least as fast as <math>\sqrt{x}</math>, this puts a rather tight bound on its rate of growth. Here, ''O'' refers to [[big O notation]].

The true rate of growth of ''M''(''x'') is not known. An unpublished conjecture of Steve Gonek states that

: <math>0 < \limsup_{x \to \infty} \frac{|M(x)|}{\sqrt{x} (\log \log \log x)^{5/4}} < \infty.</math>

Probabilistic evidence towards this conjecture is given by Nathan Ng.<ref>{{cite arXiv|title=The distribution of the summatory function of the Mobius function|author=Nathan Ng|date=October 25, 2018|eprint=math/0310381 }}</ref> In particular, Ng gives a conditional proof that the function <math>e^{-y/2} M(e^y)</math> has a limiting distribution <math>\nu</math> on <math>\mathbb{R}</math>. That is, for all bounded [[Lipschitz continuous]] functions <math>f</math> on the reals we have that

: <math>\lim_{Y\to\infty} \frac{1}{Y} \int_0^Y f\big(e^{-y/2} M(e^y)\big) \,dy = \int_{-\infty}^\infty f(x) \,d\nu(x),</math>

if one assumes various conjectures about the [[Riemann zeta function]].

==Representations==
===As an integral===
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} \int_{c-i\infty}^{c+i\infty} \frac{x^s}{s \zeta(s)} \, ds = M(x),</math>

where ''c'' > 1.

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>\operatorname{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>

Assuming that the Riemann zeta function has no multiple non-trivial zeros, one has the "exact formula" by the [[residue theorem]]:

: <math>M(x) = \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''&nbsp;=&nbsp;0.

There is also a trace formula involving a sum over the Möbius function and zeros of the Riemann zeta function in the form

: <math>\sum_{n=1}^\infty \frac{\mu(n)}{\sqrt{n}} g(\log n) = \sum_\gamma \frac{h(\gamma)}{\zeta'(1/2 + i\gamma)} + 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 the first sum on the right-hand side is taken over the non-trivial zeros of the Riemann zeta function, and (''g'',&nbsp;''h'') are related by the [[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) = -1 + \sum_{a\in\mathcal{F}_n} e^{2\pi i a},</math>

where <math>\mathcal{F}_n</math> 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 [[Logical matrix|(0,&nbsp;1) matrix]] in which
''a''<sub>''ij''</sub> is 1 if either ''j'' is 1 or ''i'' divides ''j''.

===As a sum of the number of points under ''n''-dimensional hyperboloids===
: <math>M(x) = 1 - \sum_{2 \leq a \leq x} 1 + \underset{ab \leq x}{\sum_{a \geq 2} \sum_{b \geq 2}} 1 - \underset{abc \leq x}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2}} 1 + \underset{abcd \leq x}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2} \sum_{d \geq 2}} 1 - \cdots</math>

This formulation{{citation needed|date=July 2018}} expanding the Mertens function suggests asymptotic bounds obtained by considering the [[Divisor summatory function#Piltz divisor problem|Piltz divisor problem]], which generalizes the [[Divisor summatory function|Dirichlet divisor problem]] of computing [[asymptotic estimate]]s for the summatory function of the [[divisor function]].

== Other properties ==
From <ref>{{cite journal |last1=Lehman | first1=R.S. | title= On Liouville's Function | journal= Math. Comput. |volume=14 |pages=311–320 | date= 1960}}</ref> we have 
: <math>\sum_{d=1}^{n} M(\lfloor n/d \rfloor) = 1\ .</math>

Furthermore, from <ref>{{cite journal |last1=Kanemitsu | first1=S. | last2=Yoshimoto| first2=M. | title= Farey series and the Riemann hypothesis | journal= Acta Arithmetica | date= 1996| volume=75 | issue=4 | pages=351–374 | doi=10.4064/aa-75-4-351-374 | doi-access=free }}</ref>
: <math>\sum_{d=1}^{n} M(\lfloor n/d \rfloor)d = \Phi(n)\ ,</math>
where <math>\Phi(n)</math> is the [[totient summatory function]].

== 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 all integers up to an increasing range of ''x''.<ref>{{cite journal |last1=Kotnik |first1=Tadej |last2=van de Lune |first2=Jan |title=Further systematic computations on the summatory function of the Möbius function |journal=Modelling, Analysis and Simulation |id=MAS-R0313|url=https://ir.cwi.nl/pub/4116 |date=November 2003}}</ref><ref>{{cite arXiv |last=Hurst |first=Greg |date=2016 |title=Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture |eprint=1610.08551 |class=math.NT}}</ref>

{| class="wikitable"
! Person !! Year !! Limit
|-
| Mertens || 1897 || 10<sup>4</sup>
|-
| von Sterneck || 1897 || 1.5{{e|5}}
|-
| von Sterneck || 1901 || 5{{e|5}}
|-
| von Sterneck || 1912 || 5{{e|6}}
|-
| Neubauer || 1963 || 10<sup>8</sup>
|-
| Cohen and Dress || 1979 || 7.8{{e|9}}
|-
| Dress || 1993 || 10<sup>12</sup>
|-
| Lioen and van de Lune || 1994 || 10<sup>13</sup>
|-
| Kotnik and van de Lune || 2003 || 10<sup>14</sup>
|-
| Hurst || 2016 || 10<sup>16</sup>
|}

The Mertens function for all integer values up to ''x'' may be computed in {{nobr|''O''(''x'' log log ''x'')}} time. A combinatorial algorithm has been developed incrementally starting in 1870 by [[Ernst Meissel]],<ref>{{Cite journal |last1=Meissel |first1=Ernst |date=1870 |title=Ueber die Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen |url=https://eudml.org/doc/156468 |journal=Mathematische Annalen |language=de |volume=2 |issue=4 |pages=636–642 |doi=10.1007/BF01444045 |s2cid=119828499 |issn=0025-5831}}</ref> [[Derrick Henry Lehmer|Lehmer]],<ref>{{cite journal |last=Lehmer |first=Derrick Henry |date=April 1, 1958 |title=ON THE EXACT NUMBER OF PRIMES LESS THAN A GIVEN LIMIT |url=https://projecteuclid.org/download/pdf_1/euclid.ijm/1255455259 |journal=Illinois J. Math. |volume=3 |issue=3 |pages=381–388 |access-date=February 1, 2017 }}</ref> [[Jeffrey Lagarias|Lagarias]]-[[Victor S. Miller|Miller]]-[[Andrew Odlyzko|Odlyzko]],<ref>{{cite journal |last1=Lagarias |first1=Jeffrey |last2=Miller |first2=Victor |last3 =Odlyzko |first3=Andrew |date=April 11, 1985 |title=Computing <math>\pi(x)</math>: The Meissel–Lehmer method |url=https://www.ams.org/mcom/1985-44-170/S0025-5718-1985-0777285-5/S0025-5718-1985-0777285-5.pdf |journal=Mathematics of Computation |volume=44 |issue=170 |pages=537–560 |doi=10.1090/S0025-5718-1985-0777285-5 |access-date=September 13, 2016 |doi-access=free }}</ref> and Deléglise-Rivat<ref>{{Cite journal |last1=Rivat |first1=Joöl |last2=Deléglise |first2=Marc |date=1996 |title=Computing the summation of the Möbius function |url=https://projecteuclid.org/euclid.em/1047565447 |journal=Experimental Mathematics |language=en |volume=5 |issue=4 |pages=291–295 |doi=10.1080/10586458.1996.10504594 |s2cid=574146 |issn=1944-950X}}</ref> that computes isolated values of ''M''(''x'') in {{nobr|''O''(''x''<sup>2/3</sup>(log log ''x'')<sup>1/3</sup>)}} time; a further improvement by [[Harald Helfgott]] and Lola Thompson in 2021 improves this to {{nobr|''O''(''x''<sup>3/5</sup>(log ''x'')<sup>3/5+ε</sup>)}},<ref>{{Cite journal | last1=Helfgott | first1=Harald | last2=Thompson | first2=Lola | title=Summing <math>\mu(n)</math>: a faster elementary algorithm | journal=Research in Number Theory | year=2023 | volume=9 | issue=1 | page=6 | issn=2363-9555 | doi=10.1007/s40993-022-00408-8| pmid=36511765 | pmc=9731940 }}</ref> and an algorithm by Lagarias and Odlyzko based on integrals of the [[Riemann zeta function]] achieves a running time of {{nobr|''O''(''x''<sup>1/2+ε</sup>)}}.<ref>{{Cite journal | last1=Lagarias | first1=Jeffrey | last2=Odlyzko | first2=Andrew | title=Computing <math>\pi(x)</math>: An analytic method | url=https://www.sciencedirect.com/science/article/abs/pii/019667748790037X | journal=Journal of Algorithms | date=June 1987 | volume=8 | issue=2 | pages=173–191 | doi=10.1016/0196-6774(87)90037-X}}</ref>

See {{OEIS2C|A084237}} for values of ''M''(''x'') at powers of 10.

==Known upper bounds==
Ng notes that the [[Riemann hypothesis]] (RH) is equivalent to

:<math>M(x) = O\left(\sqrt{x} \exp\left(\frac{C \cdot \log x}{\log\log x}\right)\right), </math>

for some positive constant <math>C>0</math>. Other upper bounds have been obtained by Maier, Montgomery, and Soundarajan assuming the RH including

:<math>
\begin{align} 
|M(x)| & \ll \sqrt{x} \exp\left(C_2 \cdot (\log x)^{\frac{39}{61}}\right) \\ 
|M(x)| & \ll \sqrt{x} \exp\left(\sqrt{\log x} (\log\log x)^{14}\right). 
\end{align} 
</math>

Known explicit upper bounds without assuming the RH are given by:<ref name=ElMarraki95>{{cite journal |last=El Marraki |first=M. |url=http://www.numdam.org/item/JTNB_1995__7_2_407_0/|title=Fonction sommatoire de la fonction de Möbius, 3. Majorations asymptotiques effectives fortes | journal=Journal de théorie des nombres de Bordeaux| date=1995 |volume=7 |issue=2  <!-- |access-date=16 April 2023 --> }}</ref>

:<math>
\begin{align}
|M(x)| & < \frac{12590292\cdot x}{\log^{236/75}(x)},\ \text{ for } x>\exp(12282.3) \\ 
|M(x)| & < \frac{0.6437752 \cdot x}{\log x},\ \text{ for } x>1 . 
\end{align} 
</math>

It is possible to simplify the above expression into a less restrictive but illustrative form as:

:<math>
\begin{align}
M(x)= O \left( \frac{x}{\log^{\pi}(x)} \right) . 
\end{align} 
</math>



:

==See also==
* [[Perron's formula]]
* [[Liouville's function]]

==Notes==
{{Reflist}}

== References ==
*{{cite book
  | last = Edwards
  | first = Harold | author-link = Harold Edwards (mathematician)
  | title = Riemann's Zeta Function
  | publisher = Dover
  | location = Mineola, New York
  | year = 1974
  | isbn = 0-486-41740-9}}
*{{cite journal | last1 = Mertens | first1 = F. | year = 1897 | title = "''Über eine zahlentheoretische Funktion", ''Akademie Wissenschaftlicher Wien Mathematik-Naturlich | journal = Kleine Sitzungsber, IIA | volume = 106 | pages = 761–830 }}
*{{cite journal | last1 = Odlyzko | first1 = A. M. | author-link = A. M. Odlyzko | author-link2 = Herman te Riele | last2 = te Riele | first2 = Herman | year = 1985 | title = Disproof of the Mertens Conjecture | url = http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf| journal = Journal für die reine und angewandte Mathematik | volume = 357 | pages = 138–160 }}
*{{mathworld|urlname=MertensFunction|title=Mertens function}}
*{{Cite OEIS|sequencenumber=A002321|name=Mertens's function}}
*Deléglise, M. and Rivat, J. "Computing the Summation of the Möbius Function." Experiment. Math. 5, 291-295, 1996. [https://projecteuclid.org/euclid.em/1047565447 Computing the summation of the Möbius function]
*{{cite arXiv |last=Hurst |first=Greg |date=2016 |title=Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture |eprint=1610.08551 |class=math.NT}}
*Nathan Ng, "The distribution of the summatory function of the Möbius function", Proc. London Math. Soc. (3) 89 (2004) 361-389. [http://www.cs.uleth.ca/~nathanng/RESEARCH/mobius2b.pdf]

{{DEFAULTSORT:Mertens Function}}
[[Category:Arithmetic functions]]