Dirichlet's theorem on arithmetic progressions

In number theory, Dirichlet's theorem, also called the Dirichlet prime number 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. In other words, there are infinitely many primes which are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression

$a,\ a+d,\ a+2d,\ a+3d,\ \dots,\$

and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem extends Euclid's theorem that there are infinitely many prime numbers. Stronger forms of Dirichlet's theorem state that, for any arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges, and that different arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the primes are evenly distributed (asymptotically) among each congruence class modulo d.

Note that Dirichlet's theorem does not require the prime numbers in an arithmetic sequence to be consecutive. It is also known that there exist arbitrarily long finite arithmetic progressions consisting only of primes, but this is a different result, known as the Green–Tao theorem.

Examples

An integer is a prime for the Gaussian integers if it is a prime number (in the normal sense) that is congruent to 3 modulo 4. The primes of the type 4n + 3 are

3, 7, 11, 19, 23, 31, 43, 47, 59, 67, ….

They correspond to the following values of n:

0, 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, ….

The strong form of Dirichlet's theorem implies that

$\frac{1}{3}+\frac{1}{7}+\frac{1}{11}+\frac{1}{19}+\frac{1}{23}+\frac{1}{31}+\frac{1}{43}+\frac{1}{47}+\frac{1}{59}+\frac{1}{67}+\cdots$

is a divergent series.

The following table lists several arithmetic progressions and the first few prime numbers in each of them.

Arithmetic progression	First 10 of infinitely many primes	OEIS id
2n + 1	3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …	A065091
4n + 1	5, 13, 17, 29, 37, 41, 53, 61, 73, 89, …	A002144
4n + 3	3, 7, 11, 19, 23, 31, 43, 47, 59, 67, …	A002145
6n + 1	7, 13, 19, 31, 37, 43, 61, 67, 73, 79, …	A002476
6n + 5	5, 11, 17, 23, 29, 41, 47, 53, 59, 71, …	A007528
8n + 1	17, 41, 73, 89, 97, 113, 137, 193, 233, 241, …	A007519
8n + 3	3, 11, 19, 43, 59, 67, 83, 107, 131, 139, …	A007520
8n + 5	5, 13, 29, 37, 53, 61, 101, 109, 149, 157, …	A007521
8n + 7	7, 23, 31, 47, 71, 79, 103, 127, 151, 167, …	A007522
10n + 1	11, 31, 41, 61, 71, 101, 131, 151, 181, 191, …	A030430
10n + 3	3, 13, 23, 43, 53, 73, 83, 103, 113, 163, …	A030431
10n + 7	7, 17, 37, 47, 67, 97, 107, 127, 137, 157, …	A030432
10n + 9	19, 29, 59, 79, 89, 109, 139, 149, 179, 199, …	A030433

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

$\varphi(d).\$

Further, the proportion of primes in each of those is

$\frac {1}{\varphi(d)}.\$

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

$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).

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 Legendre in his attempted unsuccessful proofs of quadratic reciprocity and proved by Dirichlet in (Dirichlet 1837) 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.

Atle Selberg (1949) gave an elementary proof.

Generalizations

The Bunyakovsky conjecture generalizes Dirichlet's theorem to higher-order polynomials. Whether or not even simple quadratic polynomials such as x² + 1 attain infinitely many prime values is an important open problem.

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

See also

• Linnik's theorem (1944)
• Bombieri–Vinogradov theorem
• Brun–Titchmarsh theorem
• Siegel–Walfisz theorem
• Dirichlet's approximation theorem

References

• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR0434929 
• Weisstein, Eric W., "Dirichlet's Theorem" from MathWorld.
• Chris Caldwell, "Dirichlet's Theorem on Primes in Arithmetic Progressions" at the Prime Pages.
• Dirichlet, P. G. L. (1837), "Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält", Abhand. Ak. Wiss. Berlin 48 
• Selberg, Atle (1949), "An elementary proof of Dirichlet's theorem about primes in an arithmetic progression", Annals of Mathematics 50 (2): 297–304, doi:10.2307/1969454, JSTOR 1969454 .

External links

• Scans of the original paper in German
• Dirichlet: There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime English translation of the original paper at the arXiv
• Dirichlet's Theorem by Jay Warendorff, Wolfram Demonstrations Project.