Dirichletscher prime number theorem

The Dirichlet prime number theorem (according to PGL Dirichlet ) is a statement from the mathematical branch of number theory , which says that an arithmetic sequence contains an infinite number of prime numbers , if this is not impossible for trivial reasons.

In its simplest form, the sentence reads: Let it be a natural number and a natural number that is too prime . Then contains the arithmetic sequence ${\displaystyle m}$ ${\displaystyle a}$ ${\displaystyle m}$

${\displaystyle a,a+m,a+2m,a+3m,\ldots }$

infinitely many prime numbers. In other words: There are infinitely many prime numbers that are congruent to modulo . ${\displaystyle a}$ ${\displaystyle m}$

If and were not prime and a common factor, then each term in the sequence would be divisible by; but two different prime numbers cannot both be divisible by . Therefore the condition of coprime number of and is necessary. ${\displaystyle a}$ ${\displaystyle m}$ ${\displaystyle g>1}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle a}$ ${\displaystyle m}$

Any odd natural number has the form or with a nonnegative integer . In this special case, the Dirichlet prime number theorem states that there are infinitely many prime numbers of both forms. ${\displaystyle 4k+1}$ ${\displaystyle 4k+3}$ ${\displaystyle k}$

In relation to the decimal system , the sentence says that there are infinitely many prime numbers that end in the decimal system with a 1, a 3, a 7 and a 9. More generally, one can say: If there are two different prime numbers that end in the same sequence of digits in a number system, there are an infinite number of other prime numbers that end in this sequence of digits in this number system.

In a quantitative version, which follows, for example, from Chebotaryov's density theorem, the Dirichlet prime number theorem reads:

${\displaystyle \lim _{x\to \infty }{\frac {\pi _{m,a}(x)}{\pi (x)}}:=\lim _{x\to \infty }{\frac {\#\{p\leq x\mid p\ \mathrm {prim} ,\quad p\equiv a{\pmod {m}}\}}{\#\{p\leq x\mid p\ \mathrm {prim} \}}}={\frac {1}{\varphi (m)}}}$

with the Euler φ function . This statement means that in each of the prime remainder classes there are, modulo, in a certain sense the same number of prime numbers. ${\displaystyle m}$

Dirichlet's proof (1837, more detailed 1839) was an important step towards the establishment of analytic number theory ( Dirichlet L series , Dirichlet characters , analytical class number formula for quadratic number fields). The L-function was introduced in analogy to Euler's introduction of the zeta function in the prime number distribution. Dirichlet then showed the non-disappearance of the L-function at position 1. The conjecture about prime numbers in arithmetic sequences comes from Adrien-Marie Legendre , who gave an incorrect proof in his textbook on number theory, as Dirichlet explained.

The error term in the manner described by the set of Dirichlet distribution of prime numbers is the subject of the set of seal Walfisz , the set of Bombieri and Vinogradov and the presumption of Elliott and Halberstam .

literature

• PGL Dirichlet: Proof of the theorem that every unlimited arithmetic progression whose first term and difference are integers without a common factor contains an infinite number of prime numbers . In: Abhand. Ak. Knowledge Berlin , 48, 1837 ( bbaw.de )
  • Research on various applications of analysis à la théorie des nombres . In: Journal für Reine und Angewandte Mathematik , Volume 19, 1839, pp. 324–369, Volume 21, 1840, pp. 1–12, 134–155 (and Dirichlet, Werke, Volume 1)
• Winfried Scharlau , Hans Opolka: From Fermat to Minkowski . Springer, 1985
• Władysław Narkiewicz : The development of prime number theory . Springer, 2000

Web links

• Eric W. Weisstein : Dirichlet's Theorem . In: MathWorld (English).
• A. Granville, G. Martin: Prime Number Races . 2004, arxiv : math / 0408319
• Dirichlet's Theorem on Primes in Arithmetic Progressions