Dedekind psi function

In 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. If is a square-free number then , where is the divisor function. The function can also be defined by setting for powers of any prime , and then extending the definition to all integers by multiplicativity. This also leads to a proof of the generating function in terms of the Riemann zeta function, which is This is also a consequence of the fact that we can write as a Dirichlet convolution of . There is an additive definition of the psi function as well. Quoting from Dickson, R. Dedekind proved that, if is decomposed in every way into a product and if is the g.c.d. of then where ranges over all divisors of and over the prime divisors of and is the totient function.