Dirichlet's theorem on arithmetic progressions
Distribution

Since the primes thin out, on average, in accordance with the prime number theorem, the same must be true for the primes in arithmetic progressions. It is natural to ask about the way the primes are shared between the various arithmetic progressions for a given value of d (there are d of those, essentially, if we do not distinguish two progressions sharing almost all their terms). The answer is given in this form: the number of feasible progressions modulo d — those where a and d do not have a common factor > 1 — is given by Euler's totient function

Further, the proportion of primes in each of those is

For example, if d is a prime number q, each of the q − 1 progressions

(all except q, 2 q, 3 q, … {\displaystyle q,2q,3q,\dots \ })

contains a proportion 1/(q − 1) of the primes.

When compared to each other, progressions with a quadratic nonresidue remainder have typically slightly more elements than those with a quadratic residue remainder (Chebyshev's bias).
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