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 formula_2 dividing formula_3 (By convention, formula_4, which is the empty product, has value 1.) The function was introduced by Richard Dedekind in connection with modular functions. The value of formula_5 for the first few integers formula_6 is: The function formula_5 is greater than formula_6 for all formula_6 greater than 1, and is even for all formula_6 greater than 2. If formula_6 is a square-free number then formula_12, where formula_13 is the  divisor function. The formula_14 function can also be defined by setting formula_15 for powers of any prime formula_2, 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 formula_18. There is an additive definition of the psi function as well. Quoting from Dickson, R. Dedekind proved that, if formula_6 is decomposed in every way into a product formula_20 and if formula_21 is the g.c.d . of formula_22 then where formula_24 ranges over all divisors of formula_6 and formula_2 over the prime divisors of formula_6 and formula_28 is the totient function. Higher orders. The generalization to higher orders via ratios of Jordan's totient is with Dirichlet series It is also the Dirichlet convolution of a power and the square of the Möbius function, If is the characteristic function of the squares, another Dirichlet convolution leads to the generalized σ-function,