查看“︁梅滕斯函數”︁的源代码

<!--[[File:Mertens.svg|thumb|right|一萬以內的梅滕斯函數]]
[[File:Mertens-big.svg|thumb|right|一千萬以內的梅滕斯函數]]-->
[[File:Congettura Mertens.png|thumb|right|图示为[[梅滕斯函數|梅滕斯函数]]的前10000项与梅滕斯猜想中的界限]]

'''梅滕斯函數'''（Mertens function）為一[[數論]]中的函數，針對所有[[正整數]]''n''定义，得名自[[弗朗茨·梅滕斯]]，梅滕斯函數定义如下

:<math>M(n) = \sum_{k=1}^n \mu(k)</math>，

其中μ是[[默比乌斯函数]]。

上述定義也可以延伸到[[實數]]：

:<math>M(x) = \sum_{1\le k \le x} \mu(k).</math>

以較不嚴謹的說法來看，M(n)是計算到n為止的[[无平方数因数的数]]，其中有偶數個質因數的個數，減去有奇數個質因數的個數。

==梅滕斯函數的值及其零點==
前160個梅滕斯函數的值為{{OEIS|id=A002321}}

{|class="wikitable"
|n
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|11
|12
|13
|14
|15
|16
|17
|18
|19
|20
|-
|''M''(''n'')
|1
|0
|&#045;1
|&#045;1
|&#045;2
|&#045;1
|&#045;2
|&#045;2
|&#045;2
|&#045;1
|&#045;2
|&#045;2
|&#045;3
|&#045;2
|&#045;1
|&#045;1
|&#045;2
|&#045;2
|&#045;3
|&#045;3
|-
|n
|21
|22
|23
|24
|25
|26
|27
|28
|29
|30
|31
|32
|33
|34
|35
|36
|37
|38
|39
|40
|-
|''M''(''n'')
|&#045;2
|&#045;1
|&#045;2
|&#045;2
|&#045;2
|&#045;1
|&#045;1
|&#045;1
|&#045;2
|&#045;3
|&#045;4
|&#045;4
|&#045;3
|&#045;2
|&#045;1
|&#045;1
|&#045;2
|&#045;1
|0
|0
|-
|n
|41
|42
|43
|44
|45
|46
|47
|48
|49
|50
|51
|52
|53
|54
|55
|56
|57
|58
|59
|60
|-
|''M''(''n'')
|&#045;1
|&#045;2
|&#045;3
|&#045;3
|&#045;3
|&#045;2
|&#045;3
|&#045;3
|&#045;3
|&#045;3
|&#045;2
|&#045;2
|&#045;3
|&#045;3
|&#045;2
|&#045;2
|&#045;1
|0
|&#045;1
|&#045;1
|-
|n
|61
|62
|63
|64
|65
|66
|67
|68
|69
|70
|71
|72
|73
|74
|75
|76
|77
|78
|79
|80
|-
|''M''(''n'')
|&#045;2
|&#045;1
|&#045;1
|&#045;1
|0
|&#045;1
|&#045;2
|&#045;2
|&#045;1
|&#045;2
|&#045;3
|&#045;3
|&#045;4
|&#045;3
|&#045;3
|&#045;3
|&#045;2
|&#045;3
|&#045;4
|&#045;4
|-
|n
|81
|82
|83
|84
|85
|86
|87
|88
|89
|90
|91
|92
|93
|94
|95
|96
|97
|98
|99
|100
|-
|''M''(''n'')
|&#045;4
|&#045;3
|&#045;4
|&#045;4
|&#045;3
|&#045;2
|&#045;1
|&#045;1
|&#045;2
|&#045;2
|&#045;1
|&#045;1
|0
|1
|2
|2
|1
|1
|1
|1
|-
|n
|101
|102
|103
|104
|105
|106
|107
|108
|109
|110
|111
|112
|113
|114
|115
|116
|117
|118
|119
|120
|-
|''M''(''n'')
|0
|&#045;1
|&#045;2
|&#045;2
|&#045;3
|&#045;2
|&#045;3
|&#045;3
|&#045;4
|&#045;5
|&#045;4
|&#045;4
|&#045;5
|&#045;6
|&#045;5
|&#045;5
|&#045;5
|&#045;4
|&#045;3
|&#045;3
|-
|n
|121
|122
|123
|124
|125
|126
|127
|128
|129
|130
|131
|132
|133
|134
|135
|136
|137
|138
|139
|140
|-
|''M''(''n'')
|&#045;3
|&#045;2
|&#045;1
|&#045;1
|&#045;1
|&#045;1
|&#045;2
|&#045;2
|&#045;1
|&#045;2
|&#045;3
|&#045;3
|&#045;2
|&#045;1
|&#045;1
|&#045;1
|&#045;2
|&#045;3
|&#045;4
|&#045;4
|-
|n
|141
|142
|143
|144
|145
|146
|147
|148
|149
|150
|151
|152
|153
|154
|155
|156
|157
|158
|159
|160
|-
|''M''(''n'')
|&#045;3
|&#045;2
|&#045;1
|&#045;1
|0
|1
|1
|1
|0
|0
|&#045;1
|&#045;1
|&#045;1
|&#045;2
|&#045;1
|&#045;1
|&#045;2
|&#045;1
|0
|0
|}

梅滕斯函數緩慢的增長及減少，不論其平均值或是峰值都有類似特性，其函數以類似混沌的方式,在零的上下變化，梅滕斯函數在以下幾點的數值為零：
: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}}.

=== 實際計算 ===
利用類似質數計算的[[埃拉托斯特尼筛法]]，可以隨著''n''的增加，計算梅滕斯函數

{| cellpadding="5" border="1"
|- style="background:gray; color:white"
| 計算者 || 年份 || 上限
|- 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>
|}

所有不大於''N''正整數的梅滕斯函數可以在用O(N<sup>2/3+ε</sup>)時間內算出來，不過已知有更好的演算法。有基本的演算法可以計算單獨的''M(N)''，時間複雜度為''O(N<sup>2/3</sup>*(ln ln(N))<sup>1/3</sup>)''。

{{OEIS2C|A084237}}為[[10的冪]]下的梅滕斯函數。

==梅滕斯猜想和黎曼猜想==
因為默比乌斯函数的數值只有-1、0及+1，因此梅滕斯函數緩慢的變化，不存在正整數''n''使得|''M''(''n'')| >&nbsp;''n''。[[默滕斯猜想|梅滕斯猜想]]更進一步，認為不存在正整數''n''使得梅滕斯函數的絕對值超過數值的平方根。梅滕斯猜想是由[[汤姆斯·斯蒂尔吉斯]]在一封于1885年写给[[夏尔·埃尔米特]]与[[弗朗茨·梅滕斯]]的信中提出的，已在1985年被{{le|安德魯·奧德里茲科|Andrew Odlyzko}}與{{le|赫爾曼·特里爾|Herman te Riele}}證否<ref>{{Citation | last1=Odlyzko | first1=A. M. | last2=te Riele | first2=H. J. J. | title=Disproof of the Mertens conjecture | url=http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf | doi=10.1515/crll.1985.357.138 | mr=783538 | year=1985 | journal=Journal für die reine und angewandte Mathematik | volume=357 | pages=138–160 | zbl=0544.10047 | issn=0075-4102 | accessdate=2015-07-25 | archive-date=2015-09-12 | archive-url=https://web.archive.org/web/20150912022420/http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf | dead-url=no }}</ref>。

[[黎曼猜想]]等價於較弱型式的梅滕斯猜想''M''(''n'') = ''O''(''n''<sup>1/2 + ε</sup>)。因為較高的''M''(''n'')成長的速度至少和''n''的平方根一様快，因此可以對成長速率定出上下限。此處的''O''為[[大O符号]]。

== 参见 ==
*[[梅滕斯猜想]]
* [[劉維爾函數]]

== 參考資料 ==
{{reflist}}
*{{cite book
  | last = Edwards
  | first = Harold 
  | 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.
* {{mathworld|urlname=MertensFunction|title=Mertens function}}
* {{cite web | url=http://www.prespacetime.com/index.php/pst/issue/view/42 | title=Borel Resummation & the Solution of Integral Equations | accessdate=2015-07-26 | author=Jose Javier Garcia Moreta | archive-date=2015-07-25 | archive-url=https://web.archive.org/web/20150725210759/http://www.prespacetime.com/index.php/pst/issue/view/42 | dead-url=no }}
* {{SloanesRef |sequencenumber=A002321|name=Mertens's function}}
* {{cite journal | url=http://projecteuclid.org/euclid.em/1047565447 | title=Computing the Summation of the Möbius Function. | author=Marc Deléglise, Joöl Rivat | journal=Experiment. Math | year=1996 | volume=2 | pages=291-295 | access-date=2015-07-25 | archive-date=2015-07-26 | archive-url=https://web.archive.org/web/20150726015526/http://projecteuclid.org/euclid.em/1047565447 | dead-url=no }}

[[Category:解析数论]]