|
:''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.'' 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 ''a'' and ''b'' are coprime, then :''f''(''ab'') = ''f''(''a'') ''f''(''b''). An arithmetic function ''f''(''n'') is said to be 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. == Examples == Some multiplicative functions are defined to make formulas easier to write: * 1(''n''): the constant function, defined by 1(''n'') = 1 (completely multiplicative) * Id(''n''): identity function, defined by Id(''n'') = ''n'' (completely multiplicative) * Id''k''(''n''): the power functions, defined by Id''k''(''n'') = ''n''''k'' for any complex number ''k'' (completely multiplicative). As special cases we have * * Id0(''n'') = 1(''n'') and * * Id1(''n'') = Id(''n''). * ''ε''(''n''): the function defined by ''ε''(''n'') = 1 if ''n'' = 1 and 0 otherwise, sometimes called ''multiplication unit for Dirichlet convolution'' or simply the ''unit function'' (completely multiplicative). Sometimes written as ''u''(''n''), but not to be confused with ''μ''(''n'') . * 1''C''(''n''), the indicator function of the set ''C'' ⊂ Z, for certain sets ''C''. The indicator function 1''C''(''n'') is multiplicative precisely when the set ''C'' has the following property for any coprime numbers ''a'' and ''b'': the product ''ab'' is in ''C'' if and only if the numbers ''a'' and ''b'' are both themselves in ''C''. This is the case if ''C'' is the set of squares, cubes, or ''k''-th powers, or if ''C'' is the set of square-free numbers. Other examples of multiplicative functions include many functions of importance in number theory, such as: * gcd(''n'',''k''): the greatest common divisor of ''n'' and ''k'', as a function of ''n'', where ''k'' is a fixed integer. * (''n''): Euler's totient function , counting the positive integers coprime to (but not bigger than) ''n'' * ''μ''(''n''): the Möbius function, the parity (−1 for odd, +1 for even) of the number of prime factors of square-free numbers; 0 if ''n'' is not square-free * ''σ''''k''(''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 * * ''σ''0(''n'') = ''d''(''n'') the number of positive divisors of ''n'', * * ''σ''1(''n'') = ''σ''(''n''), the sum of all the positive divisors of ''n''. * ''a''(''n''): the number of non-isomorphic abelian groups of order ''n''. * ''λ''(''n''): the Liouville function, ''λ''(''n'') = (−1)Ω(''n'') where Ω(''n'') is the total number of primes (counted with multiplicity) dividing ''n''. (completely multiplicative). * ''γ''(''n''), defined by ''γ''(''n'') = (−1)''ω''(n), where the additive function ''ω''(''n'') is the number of distinct primes dividing ''n''. * ''τ''(''n''): the Ramanujan tau function. * All Dirichlet characters are completely multiplicative functions. For example * * (''n''/''p''), the Legendre symbol, considered as a function of ''n'' where ''p'' is a fixed prime number. An example of a non-multiplicative function is the arithmetic function ''r''''2''(''n'') - the number of representations of ''n'' as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed. For example: :1 = 12 + 02 = (-1)2 + 02 = 02 + 12 = 02 + (-1)2 and therefore ''r''2(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, ''r''2(''n'')/4 is multiplicative. In the On-Line Encyclopedia of Integer Sequences, (sequences of values of a multiplicative function ) have the keyword "mult". See arithmetic function for some other examples of non-multiplicative functions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Multiplicative function」の詳細全文を読む スポンサード リンク
|