The theorem asserting that for any arithmetic progression a, a+d, a+2d, ... with integers a and d satisfying gcd(a,d)=1, there are infinitely many prime numbers in that progression.

Principle

Coprimality of the first term and modulus ensures nontrivial multiplicative characters modulo d exist and the associated L-series do not vanish at s=1 for the principal reasons needed, producing nonzero analytic contributions that imply infinitely many primes in each coprime residue class.

Demonstration

A concrete instance: there are infinitely many primes congruent to 1 modulo 4 and infinitely many congruent to 3 modulo 4; Dirichlet's theorem guarantees this by applying characters modulo 4 and studying the corresponding L-series to show divergence of partial sums of reciprocals over primes in the progression.

Misapplication

Assuming the theorem gives effective counts for primes in small ranges or uniformity without error terms; Dirichlet guarantees infinitude but does not by itself provide precise asymptotics or short-interval distribution without deeper results (e.g., the prime number theorem for arithmetic progressions).

Consequence

Establishes that primes are distributed across coprime residue classes, underpins much of analytic and algebraic number theory (including class field theory and generalized prime distribution results), and supplies a starting point for effective and density refinements.

Reversal

If gcd(a,d)>1 then the progression either contains no primes beyond possibly a single exceptional prime equal to the common divisor; thus the theorem's conclusion fails without the coprimality hypothesis and the reversed situation is trivial or empty of primes.

Boundary

Applies to arithmetic progressions of integers with modulus d≥1 and requires gcd(a,d)=1; it concerns ordinary primes in Z and does not directly address primes in number fields, primes of special forms without coprimality, or effective error bounds without supplementary theorems.

Semantic Tension

Dirichlet's theorem guarantees infinitude but should be distinguished from density results such as the prime number theorem in progressions or from stronger distributional statements like equidistribution or Chebotarev density; confusion arises when infinitude is conflated with quantitative uniformity.

Synthesis

Dirichlet's theorem shows that coprime residue classes are nonempty of primes in an infinite sense: by analytic properties of L-series attached to characters modulo d, every arithmetic progression with first term coprime to the modulus contains infinitely many primes, establishing a fundamental regularity in prime distribution.