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Notes on analytic number theory

Kiran S. Kedlaya

Chapter 8 Error bounds in the prime number theorem

In this chapter, we set up the proof of an error bound for the prime number theorem, modulo some further analytic properties of \(\zeta\) which will be established in Chapter 9 and Chapter 10. For the corresponding discussion for the prime number theorem in arithmetic progressions, see Chapter 11.

Section 8.1 Zeta zeroes and prime numbers

Definition 8.1.

For \(x \notin \NN\text{,}\) define the counting function
\begin{equation*} \psi(x) := \sum_{n \leq x} \Lambda(n), \end{equation*}
where \(\Lambda\) is the von Mangoldt function (Exercise 3.4.6). For \(x \in \NN\text{,}\) it is convenient to modify the definition to
\begin{equation*} \psi(x) := \sum_{n \lt x} \Lambda(n) + \frac{1}{2} \Lambda(x). \end{equation*}
Note that \(\psi\) counts prime powers rather than primes, but this doesn't affect any asymptotics as we have seen already (e.g., in (2.4.1)). To summarize, for the function \(\vartheta\) we introduced in Section 2.3:
\begin{equation*} \vartheta(x) = \sum_{p \leq x} \log p, \end{equation*}
we have
\begin{equation*} \psi(x) - \vartheta(x) = O(x^{1/2} \log x) \qquad (x \to \infty). \end{equation*}
Consequently, the prime number theorem is equivalent to
\begin{equation*} \psi(x) \sim x \qquad (x \to \infty), \end{equation*}
and we will formulate error estimates for the prime number theorem in terms of \(\psi(x)-x\text{.}\) For the translation in terms of \(\pi(x)\text{,}\) see Exercise 8.4.2.
The formula of von Mangoldt expresses the difference \(\psi(x) - x\) in terms of the zeroes of \(\zeta(s)\text{.}\) We will prove this formula in Chapter 10.

Definition 8.3.

The region \(\Real(s) \in [0,1]\) is called the critical strip for \(\zeta\text{,.}\) This terminology is used because we can account for all of the zeroes outside of the critical strip: they are the trivial zeroes \(s =-2, -4, \dots\) (Remark 6.2). In fact, the term
\begin{equation*} - \frac{1}{2} \log (1-x^{-2}) \end{equation*}
in (8.1.1) is precisely \(- \sum_{\rho} \frac{x^\rho}{\rho}\) for \(\rho\) running over the trivial zeroes.

Remark 8.4.

By numerical approximating the integral representation of \(\xi(s)\) ((6.1.4)), one can make a rigorous computer calculation that verifies that there are no real zeroes of \(\zeta\) in the critical strip. Note that in general, such techniques can be used to prove that an analytic function does not vanish in a region, but not that it does vanish at a particular point. The best one can do in that direction is to use a zero-counting formula to prove that there must be a zero near a given point.
Note that for \(x\) fixed, \(R(x,T) = o(1)\) as \(T \to \infty\text{,}\) so we have
\begin{equation*} \psi(x) - x = - \sum_{\rho} \frac{x^\rho}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) \end{equation*}
as long as we interpret the sum over \(\rho\) to mean the limit of the partial sums over \(|\Imag(\rho)| \lt T\) as \(T \to \infty\text{.}\) This formula, while pretty, is not as useful in practice as the form with remainder; we will use the remainder form by taking \(T\) to be some (preferably large) function of \(x\) as \(x \to \infty\text{.}\)

Section 8.2 How to use von Mangoldt's formula

Let us now apply von Mangoldt's formula to bound \(\psi(x) - x\text{.}\) We will see from this that we need to control the nontrivial zeroes \(\rho\) of \(\zeta\) in two respects: they cannot lie too close to the edge of the critical strip, and there cannot be too many of them in a bounded region.
Put \(\beta = \Real(\rho), \gamma = \Imag(\rho)\text{.}\) Suppose we can prove that \(\beta \lt 1 - f(|\gamma|)\) for some nonincreasing function \(f\colon [0, \infty) \to (0,1/2)\text{;}\) then
\begin{equation*} |x^\rho| = x^\beta \lt x^{1 - f(|\gamma|)} \lt x^{1 - f(T)} \end{equation*}
and \(|\rho| \geq |\gamma|\text{.}\) We thus have
\begin{equation} \left| \sum_{\rho: |\gamma| \lt T} \frac{x^\rho}{\rho} \right| \leq x^{1-f(T)} \sum_{\rho: |\gamma| \lt T} \frac{1}{\gamma}.\tag{8.2.1} \end{equation}
Let \(N(T)\) be the number of zeroes in the critical strip with \(|\gamma| \leq T\text{.}\) Then
\begin{equation} \sum_{\rho: 0 \lt |\gamma| \lt T} \frac{1}{\gamma} = \int_0^T t^{-1} dN(t) = \frac{N(T)}{T} + \int_0^T t^{-2} N(t)\,dt.\tag{8.2.2} \end{equation}
We can thus combine (8.2.1) and (8.2.2) with Theorem 8.2 to obtain a bound on \(\psi(x)-x\text{,}\) provided that we can identify suitable functions \(f\) and \(N\text{.}\) We state the relevant results here; their proofs will be given in subsequent chapters.
Putting everything together, we obtain the following. For the formulation in terms of \(\pi(x)\text{,}\) see Exercise 8.4.2.
By (8.2.2) and Theorem 8.6,
\begin{equation*} \left| \sum_{\rho: |\gamma| \lt T} \frac{1}{\gamma} \right|= O(\log^2 T). \end{equation*}
Combining this with (8.2.1) yields
\begin{equation*} \left| \sum_{\rho: |\gamma| \lt T} \frac{x^\rho}{\rho} \right| = O(x^{1-f(T)} \log^2 T) \end{equation*}
where by Theorem 8.5 we may take \(f(T) = c/(\log T)\text{.}\) Further combining with (8.1.1) yields
\begin{equation*} |\psi(x) - x| = O\left( \frac{x \log^2(xT)}{T} + x (\log T)^2 \exp(-c (\log x)/(\log T)). \end{equation*}
To balance the two estimates, we take \(T = \exp(\sqrt{\log x})\text{;}\) this yields the claimed bound.

Section 8.3 The Riemann Hypothesis

Riemann calculated a few of the zeroes of \(\zeta\) (by hand of course) and, based on this evidence, made the following remarkable conjecture.
Conjecture 8.8 has now been verified numerically for literally trillions of zeroes, e.g., see [20]. However, a proof seems sufficiently elusive that the Clay Mathematics Institute has announced a prize of US$1,000,000 for it.
This is a best-case scenario in terms of deducing error bounds on \(\psi(x) - x\text{.}\) Namely, suppose every nontrivial zero \(\rho\) of \(\zeta\) satisfies \(c \leq \Real(\rho) \leq 1-c\) for some \(c \in (0,1/2)\text{;}\) then we can take \(f(T) = c\) in (8.2.1), yielding
\begin{equation*} \psi(x) - x = O\left( x^{1-c}\log^2 T(x) + \frac{x \log^2 x}{T(x)} + \frac{x \log^2 T(x)}{T(x)} \right). \end{equation*}
By taking \(T(x) = x\text{,}\) we obtain
\begin{equation*} \psi(x) - x = O( x^{1-c} \log^2 x). \end{equation*}
If we assume Conjecture 8.8, this improves to
\begin{equation} \psi(x) - x = O(x^{1/2 + \epsilon}) \qquad (\epsilon > 0),\tag{8.3.1} \end{equation}
and similarly for \(\pi(x)\) (see Exercise 8.4.1).
Unfortunately, existing results in the direction of Conjecture 8.8 are not much stronger than Theorem 8.5. In particular, for no value of \(c>0\) are we able at present to prove that every nontrivial zero \(\rho\) satisfies \(\Real(\rho) \leq 1-c\text{.}\)

Exercises 8.4 Exercises

1.

Assume that \(\psi(x) = x + o(x^{1-\epsilon})\) for some given \(\epsilon \in (0,1/2)\text{.}\) Deduce a corresponding upper bound for \(\pi(x) - \li(x)\text{,}\) where \(\li(x)\) is the logarithmic integral function
\begin{equation*} \li(x) = \int_2^x \frac{dt}{\log t}. \end{equation*}
Then deduce that
\begin{equation*} \pi(x) - \frac{x}{\log x} \neq o(x^{1-\delta}) \end{equation*}
for any \(\delta > 0\text{.}\) (This last statement can be proved unconditionally, but don't worry about that for now.) This is the sense in which \(\li(x)\) is a better approximation than \(x/(\log x)\) of the count of primes.

2.

Deduce from Theorem 8.7 that for some \(c>0\text{,}\)
\begin{equation*} \pi(x) = \li(x) + O(x \exp(-c \sqrt{\log x})). \end{equation*}
(For comparison, we can interpret the leading term as \(x \exp(-\log \log x)\text{.}\))
Hint.
Use partial summation.
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