Theorem 7.0.1. Dirichlet’s Theorem.
For each \(a \in (\bbZ/q\bbZ)^\times\text{,}\)
\begin{equation}
\sum_{p \equiv a \pmod{q}, p \leq x} \frac{\log p}{p} = \frac{1}{\phi(q)} \log x + O(1)\text{.}\tag{7.0.3}
\end{equation}
Chapter 7 Dirichlet’s Theorem
An arithmetic progression is a sequence of numbers \(a+kq\text{,}\) \(k \geq 0\text{,}\) where \(a\) and \(q\) are fixed integers, which for simplicity we assume are strictly positive. It is equivalent to the part of the congruence class of \(a\) modulo \(q\) which is greater than or equal to \(a\text{.}\) If \(a\) and \(q\) share a common factor, then there is at most one prime in the arithmetic progression, namely \(a\) itself if it happens to be prime.
So, fix some \(q \gt 0\) and consider the congruence classes \([0], [1], \dotsc, [q-1]\) modulo \(q\text{.}\) There are only finitely many primes in congruence classes that share a common factor with \(q\text{.}\) The remaining infinitely many primes are in congruence classes that are coprime to \(q\text{.}\) There are \(\phi(q)\) of these classes, where \(\phi\) is the Euler totient function. Recalling that these classes coprime to \(q\) are precisely the ones that have multiplicative inverses modulo \(q\text{,}\) thus corresponding to units in the ring \(\bbZ/q\bbZ\text{,}\) we may denote the set of congruence classes coprime to \(q\) as \((\bbZ/q\bbZ)^\times\text{.}\)
By the pigeonhole principle, some class coprime to \(q\) includes infinitely many primes. Dirichlet’s theorem gives the result that in fact every class coprime to \(q\) contains infinitely many primes. More precisely, if \(\gcd(a,q)=1\) then the sum \(\sum_{p \equiv a \pmod{q}} \frac{\log p}{p}\) diverges.
We will be even more precise than that. Recall that by Mertens’s theorem, \(\sum_{p \leq x} \frac{\log p}{p} = \log x + O(1)\text{.}\) Therefore
\begin{equation}
\sum_{a \in \bbZ/q\bbZ} \left( \sum_{p \equiv a \pmod{q}, p \leq x} \frac{\log p}{p} \right)
= \sum_{p \leq x} \frac{\log p}{p} = \log x + O(1)\text{.}\tag{7.0.1}
\end{equation}
There are only finitely many primes in classes that share a common factor with \(q\text{;}\) dropping those classes changes the sum by a bounded amount. Therefore,
\begin{equation}
\sum_{a \in (\bbZ/q\bbZ)^\times} \left( \sum_{p \equiv a \pmod{q}, p \leq x} \frac{\log p}{p} \right)
= \log x + O(1)\text{.}\tag{7.0.2}
\end{equation}
What we will show is that each of the inner sums, for each \(a \in (\bbZ/q\bbZ)^\times\text{,}\) contributes an equal amount to the total:
Remark 7.0.2.
What about arithmetic progressions that start at a higher value, for example \(53 + 5k\) instead of \(3 + 5k\text{?}\) The difference between the primes in these two arithmetic progressions is some finite set of primes, so the difference in the summations is some bounded amount, which simply gets absorbed in the \(O(1)\) term.
Remark 7.0.3.
One might ask other questions, for example:
- What about the sum \(\sum_{p \equiv a \pmod{q}, p \leq x} \frac{1}{p}\text{?}\) Is it \(\frac{1}{\phi(q)} \log \log x + \frac{1}{\phi(q)} b + O\left(\frac{1}{\log x}\right)\text{?}\)
- Extending the prime counting function \(\pi(x)\) (the number of primes less than or equal to \(x\)), let \(\pi(x,q,a)\) be the number of primes \(p \leq x\) such that \(p \equiv a \pmod{q}\text{.}\) The Prime Number Theorem says that \(\pi(x) \sim \frac{x}{\log x}\text{.}\) Is \(\pi(x,q,a) \sim \frac{1}{\phi(q)} \frac{x}{\log x}\text{?}\)
Both questions are answered affirmatively (“yes”) later in the textbook. See Corollary 7.25 (in section 7.5, after the proof of Dirichlet’s theorem) and see Theorem 9.19 (in section 9.7, Notes on Primes in Arithmetic Progressions).
We might get to these if time permits, otherwise we will focus on the “main” form of Dirichlet’s theorem as stated above.
