Chapter 2

Some important Dirichlet series and arithmetic functions

This chapter opens with the ``Euler product", a remarkable identity that unlocks a whole vista of further ideas. Stated in modern terms , it says that for a completely multiplicative function $a(n)$,

$$\sum_{n=1}^\infty a(n) = \prod_{p \in P} [1 - a(p)]^{-1}.$$

In other words, an infinite series is equated to a product involving only the primes. This identity encapsulates in analytic form the fact that integers are uniquely expressible as products of primes. At a stroke, applied to $a(n) = n^{-s}$, it relates the zeta function to prime numbers. It is also a fertile source of further Dirichlet series and associated convolutions. In particular, it leads to the important Dirichlet series

$$\frac{1}{\zeta (s)} = \sum_{n=1}^\infty \frac{\mu (n)}{n^s}, ... ...'(s)}{\zeta (s)} = -\sum_{n=1}^\infty \frac{\Lambda (n)}{n^s},$$

in which $\mu$ is the ``Möbius function" and $\Lambda$ is the ``von Mangoldt" function. One then checks easily that these functions satisfy the convolution identities  $\mu *u = e_1$  and  $\Lambda *u = \ell$. The definitions of these functions seem a little strange if presented without motivation, but the Euler product leads us to them in a natural way.

The summation function of $\Lambda (n)$ (denoted by $\psi (x)$) is similar to Chebyshev's $\theta (x)$, but differs by counting powers of primes as well as primes. We show that it would do just as well for the purpose of proving the prime number theorem. The final step, which must wait until chapter 3, will be to deduce the required estimation of $\psi (x)$ from the behaviour of the function $\zeta'(s)/\zeta (s)$.

Meanwhile, the methods of the present chapter are enough to give very satisfactory estimates of the summation functions of various other arithmetic functions, and of sums like  $\sum_{p \in P[x]} 1/p$. A number of results of this sort are described in sections 2.5 and 2.6. The reader wanting a fast track to the prime number theorem can defer these sections.