Primes in arithmetic progressions

For integers $q\geq2$ and $a$ with $\gcd(a,q)=1$, let $\pi(x;q,a)$ count primes at most $x$ congruent to $a$ modulo $q$. Dirichlet’s theorem states that every such progression contains infinitely many primes. The coprimality condition is structural: if a class has a fixed common divisor with the modulus, every integer in it shares that divisor. Only primes dividing the modulus can then occur, giving at most finitely many exceptional primes rather than an infinite progression of primes.

Infinitude says nothing by itself about relative frequency or the size of the first prime. The prime number theorem for arithmetic progressions is stronger: for a fixed eligible class, the count is asymptotic to $\operatorname{Li}(x)/\varphi(q)$, equivalently to $x/(\varphi(q)\log x)$. Here the logarithmic integral can be taken from two to $x$; any fixed lower-limit convention changes only a bounded constant. Asymptotic equivalence means the ratio tends to one, not that finite counts become exactly equal.

For a fixed unit class $a$, sum over all characters modulo $q$ the value at an integer multiplied by the conjugate character value at $a$. On units, finite-group orthogonality makes this sum equal to the group order when the two classes agree and zero otherwise. On nonunits every character vanishes, which agrees with the desired indicator because an eligible class contains only units. The complex conjugate is essential: omitting it generally selects an inverse class rather than the stated class.

Multiply that exact indicator by von Mangoldt weights and sum through a finite endpoint. This expresses the weighted count $\psi(x;q,a)$ as an average of the twisted counts $\psi(x,\chi)=\sum_{n\leq x}\Lambda(n)\chi(n)$. The principal character produces the main term, with the finitely many prime divisors of the modulus excluded. Nonprincipal terms are controlled through their L-functions. Nonvanishing on the line of real part one, together with an appropriate analytic argument, yields a remainder small compared with the endpoint for each fixed modulus.

A theorem for every fixed modulus does not automatically remain true when the modulus grows with the endpoint. Constants and the threshold after which an error is small may depend on that fixed modulus. Uniform results must specify a range and how the error depends on both variables. For example, the classical Siegel–Walfisz theorem provides uniform asymptotics when $q\leq(\log x)^A$ for each fixed positive $A$, with constants that may depend on that parameter and with ineffective constants in the strongest usual formulation.

Real zeros near one explain why uniform error estimates require special care. If a real character has such a zero, its explicit formula contains a slowly decaying term of the form $-x^\beta/\beta$, and orthogonality transfers that term to residue counts with a character-dependent sign. This describes a conditional mechanism, not an assertion that an exceptional zero exists. Partial summation converts logarithmically weighted prime counts to ordinary counts; contributions from higher prime powers must also be bounded. Neither step licenses replacing a quantified theorem by a claim of perfect finite balance.