Dirichlet's theorem on arithmetic progressions


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.

Contents

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, … OEISA065091
4n + 1 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, … OEISA002144
4n + 3 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, … OEISA002145
6n + 1 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, … OEISA002476
6n + 5 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, … OEISA007528
8n + 1 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, … OEISA007519
8n + 3 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, … OEISA007520
8n + 5 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, … OEISA007521
8n + 7 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, … OEISA007522
10n + 1 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, … OEISA030430
10n + 3 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, … OEISA030431
10n + 7 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, … OEISA030432
10n + 9 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, … OEISA030433

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

References

External links


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