Section 18.1 Statement of the theorem
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 (
Theorem 11.11) 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}),\tag{18.1.1}
\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-based estimate
(18.1.1). It says that if you pick out the worst error term modulo
\(N\) for
each \(N\) up to about
\(x^{1/2}\text{,}\) then add these up, you get roughly what is predicted by
(18.1.1).
Theorem 18.1. 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)^\times} \left|
\psi(x; N, m) - \frac{x}{\phi(N)} \right| \leq c x (\log x)^{-A}
\end{equation*}
for \(Q := x^{1/2} (\log x)^{-B}\text{.}\)Proof.
As remarkable as this is, it is expected that one can do better than this. That said, the following conjecture appears to be extremely hard; for instance, it is not known to follow from GRH.
Conjecture 18.2. 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)^\times} \left|
\psi(x; N, m) - \frac{x}{\phi(N)} \right| \leq c x (\log x)^{-A}
\end{equation*}
for \(Q = x^{1-\epsilon}\text{.}\)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
[13].)
Theorem 18.4. 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)^\times} \left(
\psi(x; N, m) - \frac{x}{\phi(N)} \right)^2 \leq c x^2 (\log x)^{-A}
\end{equation*}
for \(Q := x (\log x)^{-B}\text{.}\)Proof.
This will follow from Theorem 19.5.
Finally, we note that Bombieri proved a slightly stronger result which I will not be proving in this course.
Theorem 18.5. Bombieri.
For any fixed \(A>0\text{,}\) there exists \(c>0\) such that
\begin{equation*}
\sum_{N \leq Q} \max_{m \in (\ZZ/N\ZZ)^\times} \left|
\psi(x; N, m) - \frac{x}{\phi(N)} \right| \leq c x^{1/2} Q (\log x)^5
\end{equation*}
for \(x^{1/2} (\log x)^{-A} \leq Q \leq x^{1/2}\text{.}\)Proof.
See [3], \S 28 for a proof by Montgomery.
