Chapter 19 The Bombieri–Vinogradov theorem (statement)
This unit begins the third part of the course, in which we apply the results gathered in the first two parts in order to say something about the extent to which primes cluster together in short intervals.
In this unit, we state the Bombieri–Vinogradov theorem, which provides surprisingly strong control on the error terms in the prime number theorem in arithmetic progressions when we look at these error terms in aggregate. We also mention some related theorems and conjectures. To attack these (which we will do in the next unit), we will need to bring to bear everything we have studied in the course so far!
For \(m,N\) coprime positive integers, put
\begin{equation*}
\psi(x; N, m) = \sum_{n \leq x, n \equiv m \pmod{N}} \Lambda(n).
\end{equation*}
Recall that the prime number theorem in arithmetic progressions says \(\psi(x; N, m) \sim x/\phi(N)\text{,}\) and that unconditionally we could get an error term
\begin{equation*}
\psi(x; N, m) = \frac{x}{\phi(N)} + O(x (\log x)^{-A})
\end{equation*}
for any fixed \(A>0\text{.}\) This is only meaningful if \(N = O((\log x)^A)\text{.}\) However, under GRH (for the Dirichlet characters of modulus \(N\)),
\begin{equation*}
\psi(x; N, m) = \frac{x}{\phi(N)} + O(x^{1/2} (\log x)^{2}),
\end{equation*}
and this is meaningful for \(N = O(x^{1/2} (\log x)^{-2})\text{.}\)
The Bombieri–Vinogradov theorem is an amazingly strong unconditional replacement for the GRH bound. It says that if you pick out the worst error term modulo \(N\) for each \(N\) up to about \(x^{1/2}\text{,}\) and add these up, you get roughly what GRH predicts you should get.
Theorem 19.1. Bombieri–Vinogradov.
For any fixed \(A>0\text{,}\) there exist constants \(c = c(A)\) and \(B = B(A)\) such that
\begin{equation*}
\sum_{N \leq Q} \max_{m \in (\ZZ/N\ZZ)^*} \left|
\psi(x; N, m) - \frac{x}{\phi(N)} \right| \leq c x (\log x)^{-A}
\end{equation*}
for \(Q := x^{1/2} (\log x)^{-B}\text{.}\)
Conjecture 19.2. Elliott–Halberstam.
For any fixed \(A>0\) and \(\epsilon>0\text{,}\) there exists \(c>0\) such that
\begin{equation*}
\sum_{N \leq Q} \max_{m \in (\ZZ/N\ZZ)^*} \left|
\psi(x; N, m) - \frac{x}{\phi(N)} \right| \leq c x (\log x)^{-A}
\end{equation*}
for \(Q = x^{1-\epsilon}\text{.}\)
Conjecture 19.2 appears to be extremely hard; for instance, it is not known to follow from GRH. One of the results of Goldston-Pintz-Y\i ld\i r\i m is that
Conjecture 19.2 almost implies the twin primes conjecture: it implies that there are infinitely many pairs of primes at distance
\(\leq 16\text{.}\) In fact, this (with 16 replaced by some other constant, depending on
\(\epsilon\)) would follow if we could prove the weaker version of Elliott-Halberstam in which
\(Q = x^{1/2 + \epsilon}\text{,}\) for any fixed
\(\epsilon > 0\text{.}\) (Even that does not follow from GRH.)
Note that in the Bombieri–Vinogradov theorem, for each modulus
\(N\) we look at the worst error term among arithmetic progressions of that modulus. If we instead average over the progressions, we should be able to take
\(Q\) larger, and in fact that is what happens. (Note: there is a typo in the statement of the theorem in
[10].)
Theorem 19.3. Barban, Davenport, Halberstam.
For any fixed \(A>0\text{,}\) there exist constants \(c = c(A)\) and \(B = B(A)\) such that
\begin{equation*}
\sum_{N \leq Q} \sum_{m \in (\ZZ/N\ZZ)^*} \left(
\psi(x; N, m) - \frac{x}{\phi(N)} \right)^2 \leq c x^2 (\log x)^{-A}
\end{equation*}
for \(Q := x (\log x)^{-B}\text{.}\)Finally, we note that Bombieri proved a slightly stronger result, which I will not be proving in this course. (See
[2], \S 28 for a proof by Montgomery.)
Theorem 19.4.
For any fixed \(A>0\text{,}\) there exists \(c>0\) such that
\begin{equation*}
\sum_{N \leq Q} \max_{m \in (\ZZ/N\ZZ)^*} \left|
\psi(x; N, m) - \frac{x}{\phi(N)} \right| \leq c x^{1/2} Q (\log x)^5
\end{equation*}
for \(x^{1/2} (\log x)^{-A} \leq Q \leq x^{1/2}\text{.}\)\(Q = x^{1-\epsilon}\text{.}\)––
\begin{equation*}
\pi(x+y; N, m) -\pi(x; N, m) \lt \frac{2y}{\phi(N) \log(y/N)} +
O \left( \frac{y}{N \log^2 (y/N)} \right)
\end{equation*}
\(\pi(x; N,m)\)\(p \leq x\)\(p \equiv m \pmod{N}\)
\begin{equation*}
\sum_{p \leq x} \tau(p-1) \sim \frac{\zeta(2) \zeta(3)}{\zeta(6)} x,
\end{equation*}
\(\tau(n)\)\(n\text{.}\)\(\equiv 1 \pmod{d}\)\(d \leq \sqrt{x}\text{.}\)
