View source for Multiplicative arithmetic function

An [[Arithmetic function|arithmetic function]] of one argument, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654301.png" />, satisfying the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

for any pair of coprime integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654303.png" />. It is usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654304.png" /> is not identically zero (which is equivalent to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654305.png" />). A multiplicative arithmetic function is called strongly multiplicative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654306.png" /> for all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654307.png" /> and all natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654308.png" />. If (*) holds for any two numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654309.png" />, and not just for coprime numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543010.png" /> is called totally multiplicative; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543011.png" />.

Examples of multiplicative arithmetic functions. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543012.png" />, the number of natural divisors of a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543013.png" />; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543014.png" />, the sum of the natural divisors of the natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543015.png" />; the [[Euler function|Euler function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543016.png" />; and the [[Möbius function|Möbius function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543017.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543018.png" /> is a strongly-multiplicative arithmetic function, a power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543019.png" /> is a totally-multiplicative arithmetic function.



====Comments====
The convolution product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543020.png" /></td> </tr></table>

yields a [[Group|group]] structure on the multiplicative functions. The unit element is given by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543023.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543024.png" />. Another standard multiplicative function is the constant function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543027.png" />) and its inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543028.png" />, the [[Möbius function|Möbius function]]. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543030.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543031.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543033.png" />.

Formally, the [[Dirichlet series|Dirichlet series]] of a multiplicative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543034.png" /> has an [[Euler product|Euler product]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543035.png" /></td> </tr></table>

whose form simplifies considerably if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543036.png" /> is strongly or totally multiplicative.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Clarendon Press  (1960)  pp. Chapts. XVI-XVII</TD></TR></table>