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Multiplicative Number Theoretic Function

A multiplicative number theoretic function is a number theoretic function f that has the property

f(mn)=f(m)f(n)
(1)

for all pairs of relatively prime positive integers m and n.

If

n=p_1^(alpha_1)p_2^(alpha_2)...p_r^(alpha_r)
(2)

is the prime factorization of a number n, then

f(n)=f(p_1^(alpha_1))f(p_2^(alpha_2))...f(p_r^(alpha_r)).
(3)

Multiplicative number theoretic functions satisfy the amazing identity

sum_(n=1)^(infty)(f(n))/(n^s)=product_(p)[sum_(k=0)^(infty)f(p^k)p^(-ks)]
(4)
=product_(p)[1+f(p)p^(-s)+f(p^2)p^(-2s)+...],
(5)

where the product is over the primes.

See also

Multiplicative Function, Number Theoretic Function

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References

Wilf, H. Generatingfunctionology, 2nd ed. New York: Academic Press, p. 58, 1994.

Referenced on Wolfram|Alpha

Multiplicative Number Theoretic Function

Cite this as:

Weisstein, Eric W. "Multiplicative Number Theoretic Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MultiplicativeNumberTheoreticFunction.html

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