Completely multiplicative function: Difference between revisions

Revision as of 13:38, 3 August 2011

Revision as of 13:58, 3 August 2011

Line 2:

==Definition==

A '''completely multiplicative function''' (or '''totally multiplicative function''') is an [[arithmetic function]] (that is, a function whose [[Domain (mathematics)|___domain]] is the [[natural number]]s), such that ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'') ''f''(''b'') holds ''for all'' positive integers ''a'' and ''b''.<ref>{{cite book|last=Apostol|first=Tom|title=Introduction to Analytic Number Theory|year=1976|publisher=Springer|isbn=0-387-90163-9|pages=30}}</ref>

Without the requirement that ''f''(1) = 1, one could still have ''f''(1) = 0, but then ''f''(''a'') = 0 for all positive integers ''a'', so this is not a very strong restriction.

==Examples==

The easiest example of a multiplicative function is a [[monomial]] with leading coefficient 1: For any particular positive integer ''n'', define ''f''(''a'') = ''a''^''n''.

The [[Liouville function]] is a non-trivial example of a completely multiplicative function as are [[Dirichlet character]]s.

==Properties==

A completely multiplicative function is completely determined by its values at the prime numbers, a consequence of the [[fundamental theorem of arithmetic]]. Thus, if ''n'' is a product of powers of distinct primes, say ''n'' = ''p''^''a'' ''q''^''b'' ..., then ''f''(''n'') = ''f''(''p'')^''a'' ''f''(''q'')^''b'' ...

There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function ''f'' multiplicative then is completely multiplicative if and only if the [[Dirichlet inverse]] is <math>\mu f</math> where <math>\mu</math> is the [[Mobius function]].<ref>{{cite book|last=Apostol|first=Tom|title=Introduction to Analytic, Numberp. Theory|year=1976|publisher=Springer|isbn=0-387-90163-9|pages=36}}</ref>

Completely multiplicative functions also satisfy a pseudo-associative law. If ''f'' is completely multiplicative then

<math>f \cdot (g*h)=(f \cdot g)*(f \cdot h)</math>

where ''*'' represents the [[Dirichlet product]] and <math>\cdot</math> represents pointwise multiplication.<ref>Apostol pg. 49</ref>. One consequence of this is is that for any completely multiplicative function ''f'' one has

<math>f*f = \tau \cdot f.</math>

Here <math> \tau</math> is the [[divisor function]].

==See also==

*[[Dirichlet character]]

*[[Liouville function]]

==References==

<references />

[[Category:Multiplicative functions]]