Jordan's totient functionLet be a positive integer. In number theory, the Jordan's totient function of a positive integer equals the number of -tuples of positive integers that are less than or equal to and that together with form a coprime set of integers. Jordan's totient function is a generalization of Euler's totient function, which is given by . The function is named after Camille Jordan. For each , Jordan's totient function is multiplicative and may be evaluated as where ranges through the prime divisors of .
Dedekind psi functionIn number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by where the product is taken over all primes dividing (By convention, , which is the empty product, has value 1.) The function was introduced by Richard Dedekind in connection with modular functions. The value of for the first few integers is: 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... . The function is greater than for all greater than 1, and is even for all greater than 2.
Generating functionIn mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem.
