Primes in Arithmetic Progressions

Dirichlet's theorem on primes in arithmetic progressions was the first major application of L-functions to number theory. The quantitative refinement -- the prime number theorem for arithmetic progressions -- gives the asymptotic density of primes $p \equiv a \pmod{q}$ .

Dirichlet's Theorem

Definition5.6Prime Counting in Progressions

For $\gcd(a, q) = 1$ , define $\pi(x; q, a) = |\{p \leq x : p \equiv a \pmod{q}\}|$ and $\psi(x; q, a) = \sum_{\substack{n \leq x \\ n \equiv a \bmod q}} \Lambda(n)$ where $\Lambda$ is the von Mangoldt function. Using character orthogonality: $\psi(x; q, a) = \frac{1}{\phi(q)} \sum_{\chi \bmod q} \overline{\chi(a)} \psi(x, \chi)$ where $\psi(x, \chi) = \sum_{n \leq x} \chi(n)\Lambda(n)$ .

ExamplePrime Number Theorem for Progressions

The PNT for arithmetic progressions states: $\pi(x; q, a) \sim \frac{x}{\phi(q)\log x}$ as $x \to \infty$ for fixed $q$ and $\gcd(a,q) = 1$ . Equivalently, $\psi(x; q, a) \sim \frac{x}{\phi(q)}$ . The primes are equidistributed among the $\phi(q)$ reduced residue classes mod $q$ , each getting proportion $1/\phi(q)$ .

Explicit Formulas

Definition5.7Explicit Formula for $\psi(x, \chi)$

For a primitive character $\chi$ mod $q$ : $\psi(x, \chi) = \delta_{\chi = \chi_0} x - \sum_\rho \frac{x^\rho}{\rho} - \frac{L'(0, \chi)}{L(0, \chi)} - \frac{1}{2}\log(1 - x^{-2})$ where the sum runs over nontrivial zeros $\rho$ of $L(s, \chi)$ . This is the analog of the Riemann-von Mangoldt explicit formula with $\chi$ -twists. The error term in the PNT for progressions is controlled by the zero-free region of $L(s, \chi)$ for all $\chi$ mod $q$ .

Quantitative Results

RemarkBombieri-Vinogradov Theorem

The Bombieri-Vinogradov theorem provides an averaged form of GRH: $\sum_{q \leq Q} \max_{\gcd(a,q)=1} \left|\psi(x; q, a) - \frac{x}{\phi(q)}\right| \ll \frac{x}{(\log x)^A}$ for $Q \leq x^{1/2}/(\log x)^B$ and any $A > 0$ . This shows that the PNT for arithmetic progressions holds for "almost all" moduli $q \leq x^{1/2-\varepsilon}$ , which suffices for many sieve applications. The Elliott-Halberstam conjecture extends this to $Q \leq x^{1-\varepsilon}$ .

ExampleLeast Prime in a Progression

Linnik's theorem: there exists a constant $L$ such that the least prime $p \equiv a \pmod{q}$ satisfies $p \ll q^L$ . Current best: $L \leq 5$ (Xylouris, 2011). Under GRH, $L = 2 + \varepsilon$ . The actual bound is believed to be $p \ll q^{1+\varepsilon}$ , but this remains unproven.