More About the Zeta-Function

To see just how many number theoretic functions are rolled up in the zeta-function, let me list a few Dirichlet series involving zeta. There are more where these came from.
1. Remember that by definition

2. If denotes the number of positive integer divisors of n, then

Thus, the zeta-function contains information about the numbers of divisors of integers.
3. If is Möbius’ function defined by

then

4. Now we start into a series of related identities.

5. If denotes the number of distinct prime divisors of n, then

6. For our old friend the tau-function,

7. And not only that, but

8. If is Euler’s phi-function defined as the number of positive integers smaller than or equal to n and having no common factors with n, then

Thus, buried inside the zeta-function are Dirichlet generating functions for practically all the elementary number theoretic functions. And not only that, but none of these results are really very hard to prove. It’s helpful to know Möbius inversion, but the bulk of these facts can be derived by just messing about with the Dirichlet series, keeping in mind that we are always looking at series

in which is a multiplicative function, that is, in which as long as x and y have no prime factors in common. This means we can usually limit our calculations of coefficients to the case where n is a power of a prime, and these calculations are usually simple.