Linnik's theorem について

Words near each other

Dictionary Lists

mini英和辞書

翻訳と辞書　辞書検索 [ 開発暫定版 ]

スポンサードリンク

Linnik's constant ：ウィキペディア英語版

Linnik's theorem
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
:

a + nd,\

where ''n'' runs through the positive integers and ''a'' and ''d'' are any given positive coprime integers with 1 ≤ ''a'' ≤ ''d'' - 1, then:
:

p(a,d) < c d^. \;

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.
== Properties ==

It is known that ''L'' ≤ 2 for almost all integers ''d''.
On the generalized Riemann hypothesis it can be shown that
:

p(a,d) \leq (1+o(1))\varphi(d)^2 \ln^2 d \; ,

where

\varphi

is the totient function.〔
It is also conjectured that:
:

p(a,d) < d^2. \;

翻訳と辞書 : 翻訳のためのインターネットリソース