 Ramanujan defines a generalization of it, if

is the prime factorization of "n", and "s" is a complex number, let

so that "φ"1("n") = "φ"("n") is Euler's function.He proves that

and uses this to show that

Letting "s" = 1,

Note that the constant is the inverse of the one in the formula for σ("n").Von Mangoldt's function unless "n" = "pk" is a power of a prime number, in which case it is the natural logarithm log "p".