Multiplicative function: Difference between revisions

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{{short description|Function equal to the product of its values on coprime factors}}

{{hatnote|Outside number theory, the tetiplicativeterm '''multiplicative function''' is usually used for [[completely multiplicative function]]s. This article discusses number theoretic multiplicative functions.}}

In [[number theory]], a '''mumultiplicative function''' is an [[arithmetic function]] <math>f</math> of a positive [[integer]] <math>n</math> with the property that <math>f(1)=1</math> and

<math display="block">f(ab) = f(a)f(b)</math> whenever <math>a</math> and <math>b</math> are [[coprime]].

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By convention, the identity element <math>\varepsilon</math> under the Dirichlet convolution is a rational arithmetical function of order <math>(0, 0)</math>.

All rational arithmetical functions are multiplicative. A multiplicative function ''f'' is a rational arithmetical function of order <math>(r, s)</math> [[if and only if]] its Bell series is of the form

<math display="block">

{\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}=

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==Multiplicative function over {{math|''F''_''q''[''X'']}}==

Let {{math|1=''A'' = ''F''_''q''[''X'']}}, the [[polynomial ring]] over the [[finite field]] with ''q'' elements. ''A'' is a [[principal ideal ___domain]] and therefore ''A'' is a [[unique factorization ___domain]].

A complex-valued function <math>\lambda</math> on ''A'' is called '''multiplicative''' if <math>\lambda(fg)=\lambda(f)\lambda(g)</math> whenever ''f'' and ''g'' are [[relatively prime]].

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Thus it gives an estimate value of <math display="block">L_t(\tau;u) = \sum_{t=1}^T K_h(u - t/T)\begin{bmatrix} ln\tau + \frac{y^2_t}{g_t\tau} \end{bmatrix}</math>

with a local [[likelihood function]] for <math>y^2_t</math> with known <math>g_t</math> and unknown <math>\sigma^2(t/T)</math>.

== Generalizations ==