Dirichlet series totient function
A totient number is a value of Euler's totient function: that is, an m for which there is at least one n for which (n) m. The valency or multiplicity of a totient number m is the number of solutions to this equation. A nontotient is a natural number which is not a totient number. Every odd integer exceeding 1 is trivially a nontotient.Dirichlet series of the form P 1 n1 s (n)n s. To do this, a generalization of Eulers totient function is required. For a real 0 and a positive integer n, an arithmetic function (n) is de ned to be the number of positive integers mfor which gcd(m; n) 1 and 0 1, this paper establishes an identity P 1 n1 s (n)n s 1 P 1 dirichlet series totient function
Dirichlet series. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet Lfunctions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet.