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.
Bell seriesIn mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function and a prime , define the formal power series , called the Bell series of modulo as: Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem: given multiplicative functions and , one has if and only if: for all primes .
Riemann zeta functionThe Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as for , and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century.
