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
![画像:product_(p)[sum_(k=0)^(infty)f(p^k)p^(-ks)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/MultiplicativeNumberTheoreticFunction/Inline7.svg){kind=link}
![画像:product_(p)[1+f(p)p^(-s)+f(p^2)p^(-2s)+...],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/MultiplicativeNumberTheoreticFunction/Inline10.svg){kind=link}
where the product is over the primes.
See also
Multiplicative Function, Number Theoretic FunctionExplore with Wolfram|Alpha
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References
Wilf, H. Generatingfunctionology, 2nd ed. New York: Academic Press, p. 58, 1994.Referenced on Wolfram|Alpha
Multiplicative Number Theoretic FunctionCite this as:
Weisstein, Eric W. "Multiplicative Number Theoretic Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultiplicativeNumberTheoreticFunction.html