Primes in Arithmetic Progressions and other Sequences

1 § VIII.1 Dirichlet’s theorem on arithmetic progressions

• 1) For k>0 and l, integers such that (k, l)=1, the arithmetic progression kn+l, n=1, 2, …, contains infinitely many primes.

  G.L. Dirichlet. Beweis des Satzes daß jede unbegrenzte arithmetische Progression deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind unendlich viele Primzahlen enthält. Werke, Leipzig: G. Reimer, 1889, I, pp. 313–342, (Original 1837).

  Remarks.

  • (i) An elementary proof was given by Mertens.

    P. Mertens. Wiener Sitzungsb. 106 (1897), 254–282.

  • (ii) The first new “elementary proof” of Dirichlet’s theorem was published by Selberg.

    A. Selberg. An elementary proof of Dirichlet’s theorem about primes in arithmetic progression. Ann. Math. 50 (2) (1947), 297–304.

    See also

    H.N. Shapiro. On primes in arithmetic progression, (II). Ann. Math. 52 (1950), 231–243.

• 2) If k is a power of an odd prime and l is a non-residue mod k, (k, l)=1, then there exist infinitely many primes in the arithmetic...