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| :''Outside number theory, the term '''multiplicative function''' is usually used for [[completely multiplicative function]]s. This article discusses number theoretic multiplicative functions.''
| | == then added with == |
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| In [[number theory]], a '''multiplicative function''' is an [[arithmetic function]] ''f''(''n'') of the positive [[integer]] ''n'' with the property that ''f''(1) = 1 and whenever
| | 'The first day after the incident of the night, I have been thinking might go into hiding place, but then I tried to point to the site looking for inspiration, did not expect to encounter more than sin ...... he quietly went very [http://www.dmwai.com/webalizer/kate-spade-8.html マザーズバッグ ケイトスペード] attentive, from the [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード バッグ 激安] details of the scene, he After the attack imitate again, the time of the attack, the attack techniques, from a different judge assailant wounds on the victim mentality, are very accurate, and he simulated the escape, went straight to stay outside the town where the killer ...... He judged the killer instinct, evidence supporting him this judgment, all from the suspect's mental state try to figure [http://www.dmwai.com/webalizer/kate-spade-10.html ケイトスペード バッグ ショルダー] out. '肖梦琪 [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペード リボン バッグ] road.<br><br>'That is what self-esteem, selfishness and the like?' Xu Pingqiu little layman, and not understand the kind of mentality<br><br>'right, his words, it counsels of a bird, both afraid to steal, they will not grab it when there are so very short drunken madness mode, an over this time, he still will be automatically retracts . prototype afraid of death, cowardly, wretched, low self-esteem ...... so even less integrated into the surrounding [http://www.dmwai.com/webalizer/kate-spade-0.html ハンドバッグ ケイトスペード] environment, and not run 'Xiaomeng Qi smiled and said, then added with:<br><br>'I |
| ''a'' and ''b'' are [[coprime]], then
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| :''f''(''ab'') = ''f''(''a'') ''f''(''b'').
| | <ul> |
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| An arithmetic function ''f''(''n'') is said to be '''[[completely multiplicative function|completely multiplicative]]''' (or '''totally multiplicative''') if ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'') ''f''(''b'') holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime.
| | <li>[http://www.mengweixs.com/plus/feedback.php?aid=69 http://www.mengweixs.com/plus/feedback.php?aid=69]</li> |
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| == Examples ==
| | <li>[http://www.bjcfjy.com/blog/?action-viewnews-itemid-9774 http://www.bjcfjy.com/blog/?action-viewnews-itemid-9774]</li> |
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| Some multiplicative functions are defined to make formulas easier to write:
| | <li>[http://www.henanyishu.com/plus/feedback.php?aid=67 http://www.henanyishu.com/plus/feedback.php?aid=67]</li> |
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| * 1(''n''): the constant function, defined by 1(''n'') = 1 (completely multiplicative)
| | </ul> |
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| * <math>1_C(n)</math> the [[indicator function]] of the set <math>C\subset \mathbb{Z}</math>. This is multiplicative if the set ''C'' has the property that if ''a'' and ''b'' are in ''C'', gcd(''a'', ''b'')=1, then ''ab'' is also in C. This is the case if ''C'' is the set of squares, cubes, or higher powers, or if ''C'' is the set of [[square-free]] numbers.
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| * Id(''n''): [[identity function]], defined by Id(''n'') = ''n'' (completely multiplicative)
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| * Id<sub>''k''</sub>(''n''): the power functions, defined by Id<sub>''k''</sub>(''n'') = ''n''<sup>''k''</sup> for any complex number ''k'' (completely multiplicative). As special cases we have
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| ** Id<sub>0</sub>(''n'') = 1(''n'') and
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| ** Id<sub>1</sub>(''n'') = Id(''n'').
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| * <math>\epsilon</math>(''n''): the function defined by <math>\epsilon</math>(''n'') = 1 if ''n'' = 1 and 0 otherwise, sometimes called ''multiplication unit for [[Dirichlet convolution]]'' or simply the ''[[unit function]]''; the Kronecker delta δ<sub>i''n''</sub>; sometimes written as ''u''(''n''), not to be confused with <math>\mu</math>(''n'') (completely multiplicative).
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| Other examples of multiplicative functions include many functions of importance in number theory, such as:
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| * gcd(''n'',''k''): the [[greatest common divisor]] of ''n'' and ''k'', as a function of ''n'', where ''k'' is a fixed integer.
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| * <math>\varphi</math>(''n''): [[Euler's totient function]] <math>\varphi</math>, counting the positive integers [[coprime]] to (but not bigger than) ''n''
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| * <math>\mu</math>(''n''): the [[Möbius function]], the parity (−1 for odd, +1 for even) of the number of prime factors of [[square-free integer|square-free]] numbers; 0 if ''n'' is not square-free
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| * <math>\sigma</math><sub>''k''</sub>(''n''): the [[divisor function]], which is the sum of the ''k''-th powers of all the positive divisors of ''n'' (where ''k'' may be any [[complex number]]). Special cases we have
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| ** <math>\sigma</math><sub>0</sub>(''n'') = ''d''(''n'') the number of positive [[divisor]]s of ''n'',
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| ** <math>\sigma</math><sub>1</sub>(''n'') = <math>\sigma</math>(''n''), the sum of all the positive divisors of ''n''.
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| * <math>a(n)</math>: the number of non-isomorphic abelian groups of order n.
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| * <math>\lambda</math>(''n''): the [[Liouville function]], λ(''n'') = (−1)<sup>Ω(''n'')</sup> where Ω(''n'') is the total number of primes (counted with multiplicity) dividing ''n''. (completely multiplicative).
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| * <math>\gamma</math>(''n''), defined by <math>\gamma</math>(''n'') = (−1)<sup><math>\omega</math>(n)</sup>, where the [[additive function]] <math>\omega</math>(''n'') is the number of distinct primes dividing ''n''.
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| * All [[Dirichlet character]]s are completely multiplicative functions. For example
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| ** (''n''/''p''), the [[Legendre symbol]], considered as a function of ''n'' where ''p'' is a fixed [[prime number]].
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| An example of a non-multiplicative function is the arithmetic function ''r''<sub>''2''</sub>(''n'') - the number of representations of ''n'' as a sum of squares of two integers, [[positive number|positive]], [[negative number|negative]], or [[0 (number)|zero]], where in counting the number of ways, reversal of order is allowed. For example:
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| :1 = 1<sup>2</sup> + 0<sup>2</sup> = (-1)<sup>2</sup> + 0<sup>2</sup> = 0<sup>2</sup> + 1<sup>2</sup> = 0<sup>2</sup> + (-1)<sup>2</sup>
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| and therefore ''r''<sub>2</sub>(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, ''r''<sub>2</sub>(''n'')/4 is multiplicative.
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| In the [[On-Line Encyclopedia of Integer Sequences]], [http://oeis.org/search?q=keyword:mult sequences of values of a multiplicative function] have the keyword "mult".
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| See [[arithmetic function]] for some other examples of non-multiplicative functions.
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| == Properties ==
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| A multiplicative function is completely determined by its values at the powers of [[prime number]]s, a consequence of the [[fundamental theorem of arithmetic]]. Thus, if ''n'' is a product of powers of distinct primes, say ''n'' = ''p''<sup>''a''</sup> ''q''<sup>''b''</sup> ..., then
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| ''f''(''n'') = ''f''(''p''<sup>''a''</sup>) ''f''(''q''<sup>''b''</sup>) ...
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| This property of multiplicative functions significantly reduces the need for computation, as in the following examples for ''n'' = 144 = 2<sup>4</sup> · 3<sup>2</sup>:
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| : d(144) = <math>\sigma</math><sub>0</sub>(144) = <math>\sigma</math><sub>0</sub>(2<sup>4</sup>)<math>\sigma</math><sub>0</sub>(3<sup>2</sup>) = (1<sup>0</sup> + 2<sup>0</sup> + 4<sup>0</sup> + 8<sup>0</sup> + 16<sup>0</sup>)(1<sup>0</sup> + 3<sup>0</sup> + 9<sup>0</sup>) = 5 · 3 = 15,
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| : <math>\sigma</math>(144) = <math>\sigma</math><sub>1</sub>(144) = <math>\sigma</math><sub>1</sub>(2<sup>4</sup>)<math>\sigma</math><sub>1</sub>(3<sup>2</sup>) = (1<sup>1</sup> + 2<sup>1</sup> + 4<sup>1</sup> + 8<sup>1</sup> + 16<sup>1</sup>)(1<sup>1</sup> + 3<sup>1</sup> + 9<sup>1</sup>) = 31 · 13 = 403, | |
| : <math>\sigma</math><sup>*</sup>(144) = <math>\sigma</math><sup>*</sup>(2<sup>4</sup>)<math>\sigma</math><sup>*</sup>(3<sup>2</sup>) = (1<sup>1</sup> + 16<sup>1</sup>)(1<sup>1</sup> + 9<sup>1</sup>) = 17 · 10 = 170. | |
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| Similarly, we have:
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| :<math>\varphi</math>(144)=<math>\varphi</math>(2<sup>4</sup>)<math>\varphi</math>(3<sup>2</sup>) = 8 · 6 = 48
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| In general, if ''f''(''n'') is a multiplicative function and ''a'', ''b'' are any two positive integers, then
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| :''f''(''a'') · ''f''(''b'') = ''f''([[greatest common divisor|gcd]](''a'',''b'')) · ''f''([[least common multiple|lcm]](''a'',''b'')). | |
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| Every completely multiplicative function is a [[homomorphism]] of [[monoid]]s and is completely determined by its restriction to the prime numbers.
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| == Convolution ==
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| If ''f'' and ''g'' are two multiplicative functions, one defines a new multiplicative function ''f'' * ''g'', the ''[[Dirichlet convolution]]'' of ''f'' and ''g'', by
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| :<math> (f \, * \, g)(n) = \sum_{d|n} f(d) \, g \left( \frac{n}{d} \right)</math>
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| where the sum extends over all positive divisors ''d'' of ''n''.
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| With this operation, the set of all multiplicative functions turns into an [[abelian group]]; the [[identity element]] is <math>\epsilon</math>. Convolution is commutative, associative, and distributive over addition.
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| Relations among the multiplicative functions discussed above include:
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| * <math>\mu</math> * 1 = <math>\epsilon</math> (the [[Möbius inversion formula]])
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| * (<math>\mu</math> * Id<sub>''k''</sub>) * Id<sub>''k''</sub> = <math>\epsilon</math> (generalized Möbius inversion)
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| * <math>\varphi</math> * 1 = Id
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| * ''d'' = 1 * 1
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| * <math>\sigma</math> = Id * 1 = <math>\varphi</math> * ''d''
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| * <math>\sigma</math><sub>''k''</sub> = Id<sub>''k''</sub> * 1
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| * Id = <math>\varphi</math> * 1 = <math>\sigma</math> * <math>\mu</math>
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| * Id<sub>''k''</sub> = <math>\sigma</math><sub>''k''</sub> * <math>\mu</math>
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| The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the [[Dirichlet ring]].
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| === Dirichlet series for some multiplicative functions ===
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| * <math>\sum_{n\ge 1} \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}</math>
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| * <math>\sum_{n\ge 1} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}</math>
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| * <math>\sum_{n\ge 1} \frac{d(n)^2}{n^s} = \frac{\zeta(s)^4}{\zeta(2s)}</math>
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| * <math>\sum_{n\ge 1} \frac{2^{\omega(n)}}{n^s} = \frac{\zeta(s)^2}{\zeta(2s)}</math>
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| More examples are shown in the article on [[Dirichlet series]].
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| ==See also==
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| * [[Euler product]]
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| * [[Bell series]]
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| * [[Lambert series]]
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| ==References==
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| * See chapter 2 of {{Apostol IANT}}
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| ==External links==
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| *[http://planetmath.org/encyclopedia/MultiplicativeFunction.html Planet Math]
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| [[Category:Multiplicative functions|*]]
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then added with
'The first day after the incident of the night, I have been thinking might go into hiding place, but then I tried to point to the site looking for inspiration, did not expect to encounter more than sin ...... he quietly went very マザーズバッグ ケイトスペード attentive, from the ケイトスペード バッグ 激安 details of the scene, he After the attack imitate again, the time of the attack, the attack techniques, from a different judge assailant wounds on the victim mentality, are very accurate, and he simulated the escape, went straight to stay outside the town where the killer ...... He judged the killer instinct, evidence supporting him this judgment, all from the suspect's mental state try to figure ケイトスペード バッグ ショルダー out. '肖梦琪 ケイトスペード リボン バッグ road.
'That is what self-esteem, selfishness and the like?' Xu Pingqiu little layman, and not understand the kind of mentality
'right, his words, it counsels of a bird, both afraid to steal, they will not grab it when there are so very short drunken madness mode, an over this time, he still will be automatically retracts . prototype afraid of death, cowardly, wretched, low self-esteem ...... so even less integrated into the surrounding ハンドバッグ ケイトスペード environment, and not run 'Xiaomeng Qi smiled and said, then added with:
'I
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