林尼克定理

林尼克定理（英語：Linnik's theorem）是解析數論中的一個定理，它回答了一個由狄利克雷定理自然推廣的問題。它聲稱，存在着正數 c 和 L 使得：如果我們用p(a,d)表示最小的素數等差數列

a+nd,\

其中 n 跑遍正整數，a 和 d 為任何的互質正整數滿足 1≤ a ≤ d -1，則：

p(a,d)<cd^{L}.\;

本定理以尤里·林尼克（英語：Yuri Linnik）的名字命名，他證明它在1944年。^[1]^[2] 雖然林尼克的證據表明 c 和 L 是可計算數，但是他沒有提供任何數值。

性質

目前已經知道， L ≤2對於幾乎所有整數d都成立.^[3]

在廣義黎曼假設成立的前提下，有，

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

這裡 $\varphi$ 是歐拉函數.^[4] 更強的上界是

p(a,d)\leq \varphi (d)^{2}\ln ^{2}d\;,

也已證實。^[5]

目前猜測：

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

^[4]

L的邊界

常數 L 稱為林尼克常數 ^[6]

下表顯示了有關該常數迄今為止取得的進展。

L ≤	證實的年份	作者
10000	1957年	潘^[7]
5448	1958年	潘
777	1965年	陳^[8]
630	1971年	朱提拉
550	1970年	朱提拉
168	1977年	陳^[9]
80	1977年	朱提拉
36	1977年	格雷厄姆^[10]
20	1981年	格雷厄姆^[11] (之前提交的陳1979年的文件)
17	1979年	陳^[12]
16	1986年	王
13.5	1989年	陳劉^[13]^[14]
8	1990年	王^[15]
5.5	1992年	希斯-布朗
5.18	2009年	吉羅里斯
5	2011	吉羅里斯

此外，在希斯-布朗的結果，常數 c 是有效的可計算數。

參考文獻

^ Linnik, Yu. V. On the least prime in an arithmetic progression I. The basic theorem. Rec. Math. (Mat. Sbornik) N.S. 1944, 15 (57): 139–178. MR 0012111.
^ Linnik, Yu. V. On the least prime in an arithmetic progression II. The Deuring-Heilbronn phenomenon. Rec. Math. (Mat. Sbornik) N.S. 1944, 15 (57): 347–368. MR 0012112.
^ Bombieri, Enrico; Friedlander, John B.; Iwaniec, Henryk. Primes in Arithmetic Progressions to Large Moduli. III. Journal of the American Mathematical Society. 1989, 2 (2): 215–224. JSTOR 1990976. MR 0976723. doi:10.2307/1990976.
^ ^4.0 ^4.1 Heath-Brown, Roger. Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression. Proc. London Math. Soc. 1992, 64 (3): 265–338. MR 1143227. doi:10.1112/plms/s3-64.2.265.
^ Lamzouri, Y.; Li, X.; Soundararajan, K. Conditional bounds for the least quadratic non-residue and related problems. Math. Comp. 2015, 84 (295): 2391–2412. arXiv:1309.3595  . doi:10.1090/S0025-5718-2015-02925-1.
^ Guy, Richard K. Unsolved problems in number theory. Problem Books in Mathematics 1 Third. New York: Springer-Verlag. 2004: 22. ISBN 978-0-387-20860-2. MR 2076335. doi:10.1007/978-0-387-26677-0.
^ Pan, Cheng Dong. On the least prime in an arithmetical progression. Sci. Record. New Series. 1957, 1: 311–313. MR 0105398.
^ Chen, Jingrun. On the least prime in an arithmetical progression. Sci. Sinica. 1965, 14: 1868–1871.
^ Chen, Jingrun. On the least prime in an arithmetical progression and two theorems concerning the zeros of Dirichlet's $L$-functions. Sci. Sinica. 1977, 20 (5): 529–562. MR 0476668.
^ Graham, Sidney West. (學位論文). 缺少或|title=為空 (幫助)
^ Graham, S. W. On Linnik's constant. Acta Arith. 1981, 39 (2): 163–179. MR 0639625. doi:10.4064/aa-39-2-163-179.
^ Chen, Jingrun. On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet's $L$-functions. II. Sci. Sinica. 1979, 22 (8): 859–889. MR 0549597.
^ Chen, Jingrun; Liu, Jian Min. On the least prime in an arithmetical progression. III. Science in China Series A: Mathematics. 1989, 32 (6): 654–673. MR 1056044.
^ Chen, Jingrun; Liu, Jian Min. On the least prime in an arithmetical progression. IV. Science in China Series A: Mathematics. 1989, 32 (7): 792–807. MR 1058000.
^ Wang, Wei. On the least prime in an arithmetical progression. Acta Mathematica Sinica. New Series. 1991, 7 (3): 279–288. MR 1141242. doi:10.1007/BF02583005.

[1] Linnik, Yu. V. On the least prime in an arithmetic progression I. The basic theorem. Rec. Math. (Mat. Sbornik) N.S. 1944, 15 (57): 139–178. MR 0012111.

[2] Linnik, Yu. V. On the least prime in an arithmetic progression II. The Deuring-Heilbronn phenomenon. Rec. Math. (Mat. Sbornik) N.S. 1944, 15 (57): 347–368. MR 0012112.

[3] Bombieri, Enrico; Friedlander, John B.; Iwaniec, Henryk. Primes in Arithmetic Progressions to Large Moduli. III. Journal of the American Mathematical Society. 1989, 2 (2): 215–224. JSTOR 1990976. MR 0976723. doi:10.2307/1990976.

[heath-brown-4] 4.0 ^4.1 Heath-Brown, Roger. Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression. Proc. London Math. Soc. 1992, 64 (3): 265–338. MR 1143227. doi:10.1112/plms/s3-64.2.265.

[LamzouriLiSoundararajan-5] Lamzouri, Y.; Li, X.; Soundararajan, K. Conditional bounds for the least quadratic non-residue and related problems. Math. Comp. 2015, 84 (295): 2391–2412. arXiv:1309.3595  . doi:10.1090/S0025-5718-2015-02925-1.

[6] Guy, Richard K. Unsolved problems in number theory. Problem Books in Mathematics 1 Third. New York: Springer-Verlag. 2004: 22. ISBN 978-0-387-20860-2. MR 2076335. doi:10.1007/978-0-387-26677-0.

[7] Pan, Cheng Dong. On the least prime in an arithmetical progression. Sci. Record. New Series. 1957, 1: 311–313. MR 0105398.

[8] Chen, Jingrun. On the least prime in an arithmetical progression. Sci. Sinica. 1965, 14: 1868–1871.

[9] Chen, Jingrun. On the least prime in an arithmetical progression and two theorems concerning the zeros of Dirichlet's $L$-functions. Sci. Sinica. 1977, 20 (5): 529–562. MR 0476668.

[10] Graham, Sidney West. (學位論文). 缺少或|title=為空 (幫助)

[11] Graham, S. W. On Linnik's constant. Acta Arith. 1981, 39 (2): 163–179. MR 0639625. doi:10.4064/aa-39-2-163-179.

[12] Chen, Jingrun. On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet's $L$-functions. II. Sci. Sinica. 1979, 22 (8): 859–889. MR 0549597.

[13] Chen, Jingrun; Liu, Jian Min. On the least prime in an arithmetical progression. III. Science in China Series A: Mathematics. 1989, 32 (6): 654–673. MR 1056044.

[14] Chen, Jingrun; Liu, Jian Min. On the least prime in an arithmetical progression. IV. Science in China Series A: Mathematics. 1989, 32 (7): 792–807. MR 1058000.

[15] Wang, Wei. On the least prime in an arithmetical progression. Acta Mathematica Sinica. New Series. 1991, 7 (3): 279–288. MR 1141242. doi:10.1007/BF02583005.

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