直至
n
≤
30000
{\displaystyle n\leq 30000}
π
(
x
;
4
,
3
)
−
π
(
x
;
4
,
1
)
{\displaystyle \pi (x;4,3)-\pi (x;4,1)}
在數論 上，切比雪夫偏差 （Chebyshev's bias）指的是說在多數時候，在給定的界限下，除4餘3的質數個數，會多於除4餘1的質數的現象。這現象最早由帕夫努季·切比雪夫 於1853年觀察到。
設
π
(
x
;
n
,
m
)
{\displaystyle \pi (x;n,m)}
x
n
k
+
m
{\displaystyle nk+m}
質數定理 在算術數列上 的推廣，有以下關係：
π
(
x
;
4
,
1
)
∼
π
(
x
;
4
,
3
)
∼
1
2
x
log
x
.
{\displaystyle \pi (x;4,1)\sim \pi (x;4,3)\sim {\frac {1}{2}}{\frac {x}{\log x}}.}
也就是說，大約一半的質數除4餘1，另外一半則是除4餘3的。因此一個合理的猜測是
π
(
x
;
4
,
1
)
>
π
(
x
;
4
,
3
)
{\displaystyle \pi (x;4,1)>\pi (x;4,3)}
π
(
x
;
4
,
3
)
>
π
(
x
;
4
,
1
)
{\displaystyle \pi (x;4,3)>\pi (x;4,1)}
π
(
x
;
4
,
3
)
>
π
(
x
;
4
,
1
)
{\displaystyle \pi (x;4,3)>\pi (x;4,1)}
x
π
(
x
;
4
,
3
)
>
π
(
x
;
4
,
1
)
{\displaystyle \pi (x;4,3)>\pi (x;4,1)}
x
π
(
x
;
4
,
3
)
=
π
(
x
;
4
,
1
)
{\displaystyle \pi (x;4,3)=\pi (x;4,1)}
π
(
x
;
4
,
1
)
>
π
(
x
;
4
,
3
)
{\displaystyle \pi (x;4,1)>\pi (x;4,3)}
x x < 26861
π
(
x
;
4
,
3
)
≥
π
(
x
;
4
,
1
)
{\displaystyle \pi (x;4,3)\geq \pi (x;4,1)}
更一般地，若
0
<
a
,
b
<
n
{\displaystyle 0<a,b<n}
gcd
(
a
,
n
)
=
gcd
(
b
,
n
)
=
1
{\displaystyle \gcd(a,n)=\gcd(b,n)=1}
a n a n
π
(
x
;
n
,
b
)
>
π
(
x
;
n
,
a
)
{\displaystyle \pi (x;n,b)>\pi (x;n,a)}
黎曼猜想 成立的狀況下得證。
Knapowski 圖蘭 兩氏曾提出更強的猜想，認為使得
π
(
x
;
4
,
3
)
>
π
(
x
;
4
,
1
)
{\displaystyle \pi (x;4,3)>\pi (x;4,1)}
x 密度 為一，也就是說
π
(
x
;
4
,
3
)
>
π
(
x
;
4
,
1
)
{\displaystyle \pi (x;4,3)>\pi (x;4,1)}
幾乎所有 的x 對數密度 大約為0.9959.... [ 1]
相關的問題可轉化成對於
k
=
−
4
{\displaystyle k=-4}
∑
q
≤
p
,
q
is prime
(
k
q
)
>
0
{\displaystyle \sum _{q\leq p,\ q\ {\text{is prime}}}\left({\frac {k}{q}}\right)>0}
p
(
m
n
)
{\displaystyle \left({\frac {m}{n}}\right)}
克羅內克符號 ；然而對於任意給定的不為零的k
k
=
−
4
{\displaystyle k=-4}
p k p
對於正整數
k
=
1
,
2
,
3
,
⋯
{\displaystyle k=1,2,3,\cdots }
p
2, 11100143, 61981, 3, 2082927221, 5, 2, 11100143, 2, 3, 577, 61463, 2083, 11, 2, 3, 2, 11100121, 5, 2082927199, 1217, 3, 2, 5, 2, 17, 61981, 3, 719, 7, 2, 11100143, 2, 3, 23, 5, 11, 31, 2, 3, 2, 13, 17, 7, 2082927199, 3, 2, 61463, 2, 11100121, 7, 3, 17, 5, 2, 11, 2, 3, 31, 7, 5, 41, 2, 3, ... （A306499 A003658
k
=
1
,
5
,
8
,
12
,
13
,
17
,
21
,
24
,
28
,
29
,
33
,
37
,
40
,
41
,
44
,
53
,
56
,
57
,
60
,
61
,
⋯
{\displaystyle k=1,5,8,12,13,17,21,24,28,29,33,37,40,41,44,53,56,57,60,61,\cdots }
對於負整數
k
=
1
,
2
,
3
,
⋯
{\displaystyle k=1,2,3,\cdots }
p
2, 3, 608981813029, 26861, 7, 5, 2, 3, 2, 11, 5, 608981813017, 19, 3, 2, 26861, 2, 643, 11, 3, 11, 31, 2, 5, 2, 3, 608981813029, 48731, 5, 13, 2, 3, 2, 7, 11, 5, 199, 3, 2, 11, 2, 29, 53, 3, 109, 41, 2, 608981813017, 2, 3, 13, 17, 23, 5, 2, 3, 2, 1019, 5, 263, 11, 3, 2, 26861, ... （A306500 A003657
k
=
−
3
,
−
4
,
−
7
,
−
8
,
−
11
,
−
15
,
−
19
,
−
20
,
−
23
,
−
24
,
−
31
,
−
35
,
−
39
,
−
40
,
−
43
,
−
47
,
−
51
,
−
52
,
−
55
,
−
56
,
−
59
,
⋯
{\displaystyle k=-3,-4,-7,-8,-11,-15,-19,-20,-23,-24,-31,-35,-39,-40,-43,-47,-51,-52,-55,-56,-59,\cdots }
對於任意非平方整數k
(
k
p
)
=
−
1
{\displaystyle \left({\frac {k}{p}}\right)=-1}
p
(
k
p
)
=
1
{\displaystyle \left({\frac {k}{p}}\right)=1}
p
設m n
f
(
m
,
n
)
=
∑
p
is prime,
p
∣
φ
(
n
)
,
x
p
≡
m
(
mod
n
)
has a solution
(
1
p
)
{\displaystyle f(m,n)=\sum _{p{\text{ is prime, }}p\,\mid \,\varphi (n),\ x^{p}\,\equiv \,m{\pmod {n}}{\text{ has a solution }}}\left({\frac {1}{p}}\right)}
其中
φ
{\displaystyle \varphi }
歐拉函數 。
這函數的一些數值如次：
f
(
1
,
5
)
=
f
(
4
,
5
)
=
1
/
2
,
f
(
2
,
5
)
=
f
(
3
,
5
)
=
0
,
f
(
1
,
6
)
=
1
/
2
,
f
(
5
,
6
)
=
0
,
f
(
1
,
7
)
=
5
/
6
,
f
(
2
,
7
)
=
f
(
4
,
7
)
=
1
/
2
,
f
(
3
,
7
)
=
f
(
5
,
7
)
=
0
,
f
(
6
,
7
)
=
1
/
3
,
f
(
1
,
8
)
=
1
/
2
,
f
(
3
,
8
)
=
f
(
5
,
8
)
=
f
(
7
,
8
)
=
0
,
f
(
1
,
9
)
=
5
/
6
,
f
(
2
,
9
)
=
f
(
5
,
9
)
=
0
,
f
(
4
,
9
)
=
f
(
7
,
9
)
=
1
/
2
,
f
(
8
,
9
)
=
1
/
3
{\displaystyle f(1,5)=f(4,5)=1/2,f(2,5)=f(3,5)=0,f(1,6)=1/2,f(5,6)=0,f(1,7)=5/6,f(2,7)=f(4,7)=1/2,f(3,7)=f(5,7)=0,f(6,7)=1/3,f(1,8)=1/2,f(3,8)=f(5,8)=f(7,8)=0,f(1,9)=5/6,f(2,9)=f(5,9)=0,f(4,9)=f(7,9)=1/2,f(8,9)=1/3}
目前有猜想認為，若若
0
<
a
,
b
<
n
{\displaystyle 0<a,b<n}
gcd
(
a
,
n
)
=
gcd
(
b
,
n
)
=
1
{\displaystyle \gcd(a,n)=\gcd(b,n)=1}
f
(
a
,
n
)
>
f
(
b
,
n
)
{\displaystyle f(a,n)>f(b,n)}
π
(
x
;
n
,
b
)
>
π
(
x
;
n
,
a
)
{\displaystyle \pi (x;n,b)>\pi (x;n,a)}
^ (Rubinstein—Sarnak, 1994)
P.L. Chebyshev: Lettre de M. le Professeur Tchébychev à M. Fuss sur un nouveaux théorème relatif aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3, Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg , 11 (1853), 208.
Granville, Andrew ; Martin, Greg. Prime number races . Amer. Math. Monthly . 2006, 113 (1): 1–33. JSTOR 27641834 S2CID 3846453 doi:10.1080/00029890.2006.11920275 J. Kaczorowski: On the distribution of primes (mod 4), Analysis , 15 (1995), 159–171.
S. Knapowski, Turan: Comparative prime number theory, I, Acta Math. Acad. Sci. Hung. 13 (1962), 299–314.
Rubinstein, M.; Sarnak, P. Chebyshev's bias. Experimental Mathematics . 1994, 3 (3): 173–197. doi:10.1080/10586458.1994.10504289 
A306499是
A306500是
