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Dirichlet's theorem on arithmetic progressions

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In number theory, Dirichlet's theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n > 0, or in other words: there are infinitely many primes which are congruent to a modulo d.

This theorem extends Euclid's theorem that there are infinitely many primes (in this case of the form 3 + 4n, which are also the Gaussian primes, or of the form 1 + 2n, for every odd number, excluding Template:Num). Note that the theorem does not say that there are infinitely many consecutive terms in the arithmetic progression

a, a+d, a+2d, a+3d, ...,

which are prime. For example in we get primes of the type 4n + 3 'only' for n with the values

1, 2, 4, 5, 7, 10, 11, 14, 16, 17, 19, 20, 25, 26, 31, 32, 34, 37, 40, 41, 44, 47, 49, 52, 55, 56, 59, 62, 65, 67, 70, 76, 77, 82, 86, 89, 91, 94, 95, ... .

In fact since the primes thin out, on average, 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 that are not ruled out on the grounds that a and d have a common factor > 1 — is given by Euler's totient function

φ(d).

Further, the natural density of primes in each of those is

1/φ(d).

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

q, 2q, 3q, ...

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

History

Euler stated that every arithmetic progression beginning with 1 contains an infinite number of primes. The theorem in the above form was first conjectured by Gauss and proved by Dirichlet in 1835 with Dirichlet L-series. The proof is modeled on Euler's earlier work relating the Riemann zeta function to the distribution of primes. The theorem represents the beginning of rigorous analytic number theory.

In algebraic number theory Dirichlet's theorem generalizes to Chebotarev's density theorem.

See also

fr:Thorme de Dirichlet it:Teorema di Dirichlet ru:Теорема Дирихле о простых числах в арифметической прогрессии

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