Editing Multiplicative function (section)

== Generalizations ==

An arithmetical function <math>f</math> is 
quasimultiplicative if there exists a nonzero constant <math>c</math> such that
<math>
c\,f(mn)=f(m)f(n)
</math>
for all positive integers <math>m, n</math> with <math>(m, n)=1</math>. This concept originates by Lahiri (1972). 

An arithmetical function <math>f</math> is semimultiplicative 
if there exists a nonzero constant <math>c</math>, a positive integer <math>a</math> and
a multiplicative function <math>f_m</math> such that
<math>
f(n)=c f_m(n/a)
</math>
for all positive integers <math>n</math> 
(under the convention that <math>f_m(x)=0</math> if <math>x</math> is not a positive integer.) This concept is due to David Rearick (1966). 

An arithmetical function <math>f</math> is Selberg multiplicative if 
for each prime <math>p</math> there exists a function <math>f_p</math> on nonnegative integers with <math>f_p(0)=1</math> for
all but finitely many primes <math>p</math> such that
<math>
f(n)=\prod_{p} f_p(\nu_p(n))
</math>
for all positive integers <math>n</math>, where <math>\nu_p(n)</math> is the exponent of <math>p</math> in the canonical factorization of <math>n</math>. 
See Selberg (1977). 

It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide.  They both satisfy the arithmetical identity 
<math>
f(m)f(n)=f((m, n))f([m, n])
</math>
for all positive integers <math>m, n</math>. See Haukkanen (2012).

It is well known and easy to see that multiplicative functions are quasimultiplicative functions with <math>c=1</math> and quasimultiplicative functions are semimultiplicative functions with <math>a=1</math>.