Chapter 3

The basic theorems

Recall that our overall strategy is based on the series  $-\zeta'(s)/\zeta (s) = \sum_{n=1}^\infty \Lambda (n)/n^s$: the hope is to deduce the required estimation of $\psi (x)$ from the properties of the function $\zeta'(s)/\zeta (s)$. This procedure can be carried out for a general Dirichlet series: given that  $\sum_{n=1}^\infty a(n)/n^s$, converges (on a suitable domain) to a function $f(s)$, we shall show that if $f(s)$ has certain properties, then  $A(x)/x$  tends to a limit. This is our ``fundamental theorem". In this way, we finally obtain the prime number theorem as one among a family of theorems. Though it is certainly the most important one, other theorems in the family are of considerable interest; we shall meet some later in this chapter, and more in chapter 4.

The fundamental theorem will actually be obtained in three different versions, describing respectively an integral, a limit (as just mentioned) and a series. The integral version will be proved first, and the other two will be derived from it. An example of the type of series obtained is  $\sum_{n=1}^\infty \mu (n)/n = 0$.

Though we have taken care to consider the zeta function, and other Dirichlet series, as functions of a complex variable $s$, we have not really made any use of this fact so far. Complex numbers were not used in proving any of the results in chapters 1 and 2. However, the fundamental theorem depends in an essential way on the properties of $f(s)$ as a complex function. Very roughly, the method is based on the idea of ``inverting" the Dirichlet series to express $A(x)$ in terms of $f(s)$: this is done by an integral on a vertical line in the complex plane. As with power series, no such expression is available within real analysis.

The fundamental theorem requires $f(s)$ to exist, and to be holomorphic, on a region including the line Re $s = 1$ (except at the point 1). However, so far we have only defined $\zeta (s)$ for Re $s > 1$. So our first task is to find a way of extending the definition of $\zeta (s)$ to a larger domain (for example, Re $s > 0$). The extended function must be holomorphic, so that the theorems of complex analysis apply. It must also be non-zero on a region including Re $s = 1$, since the fundamental theorem is to be applied to $\zeta'(s)/\zeta (s)$, not $\zeta (s)$ itself. This extension is an interesting piece of work in its own right; it is discussed in section 3.1.

For the fundamental theorem itself, we then describe two alternative methods in sections 3.2 and 3.3. The first, a variant of the ``traditional" method, is along the lines just mentioned. The second method was devised by D.J. Newman in 1980. It is slightly shorter, but less transparent, depending on an ingeniously chosen function. It still uses complex integrals. The two versions of the theorem are not identical, because one of the hypotheses is different in the two cases. The first method needs an estimation of $\vert f(1+ it)\vert$ in terms of $t$, while Newman's method needs the condition $\vert A(x)\vert \leq Cx$ (in the case of the prime number theorem, this means Chebyshev's upper estimate). The first method is slightly preferable for the purpose of proving a more exact version of the prime number theorem with an error estimate (the subject of chapter 5). However, for the purposes of the present chapter (and indeed chapter 4), the reader may opt for either section 3.2 or section 3.3; later results are presented in a way that caters for either. Better still, read both and make comparisons!

Given the effort invested in proving the prime number theorem, one would hope to see some interesting applications. Some are described in section 3.5. For example, we show how to derive, for each $k$, an estimation of the number of integers less than $n$ having $k$ prime factors.