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Stephens' Constant
∏
p
(
1
−
p
p
3
−
1
)
=
0.57595996889294543964316337549249669...
{\displaystyle \prod _{p}\left(1-{{p} \over {p^{3}-1}}\right)=0.57595996889294543964316337549249669...}
A065478 OEIS )Stephens, P. J. to consider the generalized Riemann hypothesis , showed the density of the set of primes p.[ 1]
[ 2]
C
S
=
∏
p
(
1
−
p
p
3
−
1
)
{\displaystyle C_{S}=\prod _{p}\left(1-{{p} \over {p^{3}-1}}\right)}
=
∏
p
(
1
−
p
(
(
p
+
1
)
2
(
p
−
1
)
)
−
p
2
+
p
)
{\displaystyle \;\;\;=\prod _{p}\left(1-{{p} \over {((p+1)^{2}(p-1))-p^{2}+p}}\right)}
=
∏
p
(
(
(
p
+
1
)
2
(
p
−
1
)
)
−
p
2
+
p
−
p
(
(
p
+
1
)
2
(
p
−
1
)
)
−
p
2
+
p
)
{\displaystyle \;\;\;=\prod _{p}\left({{((p+1)^{2}(p-1))-p^{2}+p-p} \over {((p+1)^{2}(p-1))-p^{2}+p}}\right)}
=
∏
p
(
(
(
p
+
1
)
2
(
p
−
1
)
)
−
p
2
(
(
p
+
1
)
2
(
p
−
1
)
)
−
p
2
+
p
)
{\displaystyle \;\;\;=\prod _{p}\left({{((p+1)^{2}(p-1))-p^{2}} \over {((p+1)^{2}(p-1))-p^{2}+p}}\right)}
=
∏
p
(
p
3
−
p
−
1
p
3
−
1
)
{\displaystyle \;\;\;=\prod _{p}\left({{p^{3}-p-1} \over {p^{3}-1}}\right)}
=
∏
p
(
(
p
(
p
2
−
1
)
)
−
1
(
p
(
p
2
−
1
)
)
+
p
−
1
)
{\displaystyle \;\;\;=\prod _{p}\left({{(p(p^{2}-1))-1} \over {(p(p^{2}-1))+p-1}}\right)}
=
∏
p
(
(
p
(
p
2
−
1
)
)
−
1
(
(
p
+
1
)
2
(
p
−
1
)
)
−
p
2
+
p
)
{\displaystyle \;\;\;=\prod _{p}\left({{(p(p^{2}-1))-1} \over {((p+1)^{2}(p-1))-p^{2}+p}}\right)}
=
∏
p
(
(
p
(
p
2
−
1
)
)
−
1
(
(
p
+
1
)
2
(
p
−
1
)
)
−
p
(
p
−
1
)
)
{\displaystyle \;\;\;=\prod _{p}\left({{(p(p^{2}-1))-1} \over {((p+1)^{2}(p-1))-p(p-1)}}\right)}
=
∏
p
(
p
(
p
2
−
1
)
−
1
(
(
p
+
1
)
2
−
p
)
(
p
−
1
)
)
{\displaystyle \;\;\;=\prod _{p}\left({{p(p^{2}-1)-1} \over {((p+1)^{2}-p)(p-1)}}\right)}
=
∏
p
(
p
(
p
2
−
1
)
−
1
(
p
−
1
)
)
(
1
(
(
p
+
1
)
2
−
p
)
)
{\displaystyle \;\;\;=\prod _{p}\left({{p(p^{2}-1)-{1}} \over {(p-1)}}\right)\left({{1} \over {((p+1)^{2}-p)}}\right)}
=
∏
p
(
p
(
(
p
2
−
1
)
−
1
p
)
(
p
−
1
)
)
(
1
p
2
+
2
p
+
1
−
p
)
{\displaystyle \;\;\;=\prod _{p}\left({{p\left((p^{2}-1)-{{1} \over {p}}\right)} \over {(p-1)}}\right)\left({{1} \over {p^{2}+2p+1-p}}\right)}
=
∏
p
(
(
p
2
−
1
)
−
1
p
(
p
−
1
)
)
(
p
p
2
+
p
+
1
)
{\displaystyle \;\;\;=\prod _{p}\left({{(p^{2}-1)-{{1} \over {p}}} \over {(p-1)}}\right)\left({{p} \over {p^{2}+p+1}}\right)}
=
∏
p
(
(
p
2
−
1
)
−
1
p
(
p
−
1
)
)
(
p
p
(
p
+
1
+
1
p
)
)
{\displaystyle \;\;\;=\prod _{p}\left({{(p^{2}-1)-{{1} \over {p}}} \over {(p-1)}}\right)\left({{p} \over {p(p+1+{{1} \over {p}})}}\right)}
=
∏
p
(
(
p
2
−
1
)
−
1
p
p
(
p
−
1
)
)
(
p
(
p
+
1
+
1
p
)
)
{\displaystyle \;\;\;=\prod _{p}\left({{(p^{2}-1)-{{1} \over {p}}} \over {p(p-1)}}\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}
=
∏
p
(
(
(
p
2
−
1
)
p
(
p
−
1
)
)
−
(
1
p
p
(
p
−
1
)
)
)
(
p
(
p
+
1
+
1
p
)
)
{\displaystyle \;\;\;=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{{1} \over {p}} \over {p(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}
=
∏
p
(
(
(
p
2
−
1
)
p
(
p
−
1
)
)
−
(
1
p
2
(
p
−
1
)
)
)
(
p
(
p
+
1
+
1
p
)
)
{\displaystyle \;\;\;=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{1} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}
=
∏
p
(
(
(
p
2
−
1
)
p
(
p
−
1
)
)
−
(
1
(
p
−
1
)
)
+
(
p
−
1
−
p
2
(
p
−
1
)
p
2
(
p
−
1
)
2
)
)
(
p
(
p
+
1
+
1
p
)
)
{\displaystyle \;\;\;=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{1} \over {(p-1)}}\right)+\left({{p-1-p^{2}(p-1)} \over {p^{2}(p-1)^{2}}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}
=
∏
p
(
C
A
+
(
(
p
−
1
)
−
p
2
(
p
−
1
)
p
2
(
p
−
1
)
2
)
)
(
p
(
p
+
1
+
1
p
)
)
{\displaystyle \;\;\;=\prod _{p}\left(C_{A}+\left({{(p-1)-p^{2}(p-1)} \over {p^{2}(p-1)^{2}}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}
C
A
{\displaystyle C_{A}}
Artin constant
=
∏
p
(
C
A
+
(
1
−
p
2
p
2
(
p
−
1
)
)
)
(
p
(
p
+
1
+
1
p
)
)
{\displaystyle \;\;\;=\prod _{p}\left(C_{A}+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}
C
S
=
∏
p
(
(
(
p
2
−
1
)
p
(
p
−
1
)
)
−
(
1
(
p
−
1
)
)
+
(
1
−
p
2
p
2
(
p
−
1
)
)
)
(
p
(
p
+
1
+
1
p
)
)
{\displaystyle C_{S}=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{1} \over {(p-1)}}\right)+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}
C
S
=
∏
p
(
C
L
t
−
(
1
p
2
−
p
)
−
(
1
p
2
−
p
)
+
(
1
−
p
2
p
2
(
p
−
1
)
)
)
(
p
(
p
+
1
+
1
p
)
)
{\displaystyle C_{S}=\prod _{p}\left(C_{Lt}-\left({{1} \over {p^{2}-p}}\right)-\left({{1} \over {p^{2}-p}}\right)+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}
C
L
t
=
{\displaystyle C_{Lt}=}
Landau Totient constant
C
L
t
=
∏
p
(
1
+
1
p
(
p
−
1
)
)
{\displaystyle C_{Lt}=\prod _{p}\left(1+{{1} \over {p(p-1)}}\right)}
C
S
=
∏
p
(
C
L
t
−
(
1
p
2
−
p
)
−
(
1
p
2
−
p
)
+
C
t
−
(
p
p
−
1
)
)
(
p
(
p
+
1
+
1
p
)
)
{\displaystyle C_{S}=\prod _{p}\left(C_{Lt}-\left({{1} \over {p^{2}-p}}\right)-\left({{1} \over {p^{2}-p}}\right)+C_{t}-\left({{p} \over {p-1}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}
C
t
=
{\displaystyle C_{t}=}
Totient constant
C
t
=
∏
p
(
1
+
1
p
2
(
p
−
1
)
)
{\displaystyle C_{t}=\prod _{p}^{}\left(1+{{1} \over {p^{2}(p-1)}}\right)}
See also
References
^ (Stephen 1976)Stephens, P. J. "Prime Divisor of Second-Order Linear Recurrences, I." J. Number Th. 8, 313-332, 1976.
^ Mathematical Constants, Steven R. Finch -StephensConstant,with Artin's Constant(2.4)
(Binary strings without zigzags Emanuele Munarini - Norma Zagaglia Salvi,S´eminaire Lotharingien de Combinatoire 49 (2004), Article B49h)http://www.mat.univie.ac.at/~slc/wpapers/s49zagaglia.pdf (http://www.mat.univie.ac.at/~slc/wpapers/s49zagaglia.html )
