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. One naturally then asks 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 don't 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, other than
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).
Read more about this topic: Dirichlet's Theorem On Arithmetic Progressions
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