Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive c and L such that, if we denote p(a,d) the least prime in the arithmetic progression
where n runs through the positive integers and a and d are any given positive coprime integers with 1 ≤ a ≤ d − 1, then:
The theorem is named after Yuri Vladimirovich Linnik, who proved it in 1944. Although Linnik's proof showed c and L to be effectively computable, he provided no numerical values for them.
It follows from Zsigmondy's theorem that p(1,d) ≤ 2d − 1, for all d ≥ 3. It is known that p(1,p) ≤ Lp, for all primes p ≥ 5, as Lp is congruent to 1 modulo p for all prime numbers p, where Lp denotes the p-th Lucas number. Just like Mersenne numbers, Lucas numbers with prime indices have divisors of the form 2kp+1.
Properties
It is known that L ≤ 2 for almost all integers d.
On the generalized Riemann hypothesis it can be shown that
where is the totient function, and the stronger bound
has been also proved.
It is also conjectured that:
Bounds for L The constant L is called Linnik's constant and the following table shows the progress that has been made on determining its size.
L ≤ Year of publication Author 10000 1957 Pan 5448 1958 Pan 777 1965 Chen 630 1971 Jutila 550 1970 Jutila 168 1977 Chen 80 1977 Jutila 36 1977 Graham 20 1981 Graham (submitted before Chen's 1979 paper) 17 1979 Chen 16 1986 Wang 13.5 1989 Chen and Liu 8 1990 Wang 5.5 1992 Heath-Brown 5.18 2009 Xylouris 5 2011 Xylouris 5 − ε 2018 XylourisLinniks Konstante ist kleiner als 5
Moreover, in Heath-Brown's result the constant c is effectively computable.
Notes