Analytic number theory

4 Distribution of primes

Theorem 4.1 (Dirichlet’s Theorem). The polar density of primes in each relatively prime congruence class mod \(m\) are \(\frac {1}{\phi (m)}\).

Proof. Let \((a,m) = 1\), \(\real (s)>1\). \(\sum _{p \equiv a \pmod m}\frac {1}{p^s} = \frac {1}{\phi (m)}\sum _p \frac {\sum _\chi \chi (a^{-1}p)}{p^s}\) \(= \frac {1}{\phi (m)}\sum _\chi \overline \chi (a)\sum _p \frac {\chi (p)}{p^s} = \frac {1}{\phi (m)}\sum _\chi \overline \chi (a) \log (L(s,\chi )) + O(1)\). Now letting \(s\) near \(1\) and applying Corollary 3.12, it follows that all the terms in the sum are \(O(1)\) except for \(\log (L(s,1))\), which is \(\log (\zeta (s))+O(1)\), so we get \(=\frac {1}{\phi (m)}\log (\zeta (s)) + O(1)\). □

One should note that the proof above works for any cyclotomic extension of number fields without any change other than restricting to primes with residual degree \(1\) over \(\QQ \).

We can improve the results of Theorem 3.8. First suppose that \(L/K\) has cyclic Galois group of order \(n\).

Lemma 4.2. The Dirichlet density of elements of order \(d|n\) is \(\frac {\phi (d)}{n}\).

Proof. The density of elements of order dividing \(d\) is \(\frac {d}{n}\) by Theorem 3.8. But then by Möbius inversion, we are done. □

Theorem 4.3 (Frobenius Density Theorem). Let \(L\) be a Galois extension of \(K\) with Galois group \(G\), and let \(\sigma \in G\) be an element of order \(n\). Then the set of primes in \(K\) with Frobenius \(\sigma ^k\) has Dirichlet density \(c\frac {\phi (n)}{|G|}\), where \(c\) is the index of the normalizer of \(\langle \sigma \rangle \) in \(G\).

Proof. We will ignore ramifying primes, and those in \(L^\sigma \) and \(K\) with inertial degree \(>1\) over \(\QQ \). Let \(L^\sigma \) be the fixed field of \(\sigma \). By the previous lemma, the set \(A\) of primes in \(L^{\sigma }\) with Frobenius \(\sigma ^k\) has polar density \(\frac {\phi (d)}{n}\). Now let \(B\) be the set of prime in \(K\) with Frobenius \(\sigma ^k\) for some prime over them. Each prime in \(B\) has \(\frac {|G|}{n}\) primes above it in \(L\), and the Galois group acts on these transitively, which acts on the decomposition group transitively by conjugation. Thus \(\frac {|G|}{nc}\) primes above the prime in \(B\) must have a Frobenius that works. Each of these gives a different element of \(A\) that restricts to \(B\), so the restriction map from \(A\) to \(B\) is \(\frac {|G|}{nc}\) to \(1\). Looking at the level of zeta functions for \(A,B\), since everything is inertial degree \(1\) over \(\QQ \), we immediately get that the polar density of \(B\) is \(\frac {c\phi (n)}{|G|}\). □

The Chebotarev Density Theorem is a common generalization of both of the previous theorems. Here is a relatively simple approach to the Dirichlet density version of the theorem:

Theorem 4.4 (Chebotarev Density Theorem). Let \(L/K\) be a Galois extension of number fields, and let \([\sigma ]\) be a conjugacy class in the Galois group. The Dirichlet density of primes in the class \([\sigma ]\) is \(\frac {[\sigma ]}{|G|}\).

Proof. First we’ll reduce to the case of a cyclic extension using the same technique as in the previous. Given a prime \(p\) with Frobenius \([\sigma ]\), note that it splits into \(\frac {|G|}{o(\sigma )}\) primes in \(L\), and exactly \(\frac {1}{[\sigma ]}\) of those have Frobenius actually \(\sigma \). Combining this with the Dirichlet density for the cyclic case, along with ignoring primes ramifying or having nontrivial residual degree over \(\QQ \), we get the result.

Next, we will reduce to the case that \(L\) is a cyclotomic extension of \(K\), which was proven in the remark after Dirichlet’s theorem. If \(L\) is a cyclic extension, note that we only need to show that \(\frac {1}{|G|}\) is a lower bound on the lower Dirichlet density as the same lower bound on the rest of the elements of \(G\) will give the desired upper bound. This will be shown as follows: pick a prime \(m\) linearly disjoint from \(K\), and consider \(L[\zeta _m]\), whose Galois group can be identified with \(G\times \ZZ /m\ZZ ^\times \). Then if \(a \in \ZZ /m\ZZ ^\times \) is an element with \(n\) dividing its order, then \(\langle (\sigma ,a)\rangle \cap G\times \{1\}\) is a trivial group, which by Galois theory means that \(L[\zeta _m]/L[\zeta _m]^{(\sigma ,a)}\) is a cyclotomic extension, so the density of primes for \((\sigma ,a)\) is what we want. In addition, the sum of the lower Dirichlet densities for the elements \((\sigma ,a)\) as \(a\) ranges in \(\ZZ /m\ZZ ^\times \) is at most the lower density of \(\sigma \). If \(H_m\) is the number of elements elements of \(\ZZ /m\ZZ ^\times \) with \(n\) dividing its order, then the lower Dirichlet density is at least \(\frac {H_m}{(m-1)|G|}\). Now we can choose by Dirichlet’s theorem \(m \equiv 1 \pmod {n^k}\) for large \(k\) so that \(\frac {H_m}{m-1} \to 1\). □