Stephens' constant

Stephens' constant

$\prod _{p}\left(1-{\frac {p}{p^{3}-1}}\right)=0.57595996889294543964316337549249669\ldots$(sequence A065478 in the OEIS)

Stephens, P. J. to consider the generalized Riemann hypothesis, showed the density of the set of primes p.

Stephens' constant and Artin constant

${\begin{aligned}C_{S}&=\prod _{p}\left(1-{\frac {p}{p^{3}-1}}\right)\\[5pt]&=\prod _{p}\left(1-{\frac {p}{(p+1)^{2}(p-1)-p^{2}+p}}\right)\\[5pt]&=\prod _{p}\left({\frac {(p+1)^{2}(p-1)-p^{2}+p-p}{(p+1)^{2}(p-1)-p^{2}+p}}\right)\\[5pt]&=\prod _{p}\left({{((p+1)^{2}(p-1))-p^{2}} \over {((p+1)^{2}(p-1))-p^{2}+p}}\right)\\[5pt]&=\prod _{p}\left({{p^{3}-p-1} \over {p^{3}-1}}\right)\\[5pt]&=\prod _{p}\left({{(p(p^{2}-1))-1} \over {(p(p^{2}-1))+p-1}}\right)\\[5pt]&=\prod _{p}\left({{(p(p^{2}-1))-1} \over {((p+1)^{2}(p-1))-p^{2}+p}}\right)\\[5pt]&=\prod _{p}\left({{(p(p^{2}-1))-1} \over {((p+1)^{2}(p-1))-p(p-1)}}\right)\\[5pt]&=\prod _{p}\left({{p(p^{2}-1)-1} \over {((p+1)^{2}-p)(p-1)}}\right)\\[5pt]&=\prod _{p}\left({\frac {p(p^{2}-1)-1}{(p-1)}}\right)\left({\frac {1}{((p+1)^{2}-p)}}\right)\\[5pt]&=\prod _{p}\left({\frac {p\left((p^{2}-1)-{\frac {1}{p}}\right)}{(p-1)}}\right)\left({\frac {1}{p^{2}+2p+1-p}}\right)\\[5pt]&=\prod _{p}\left({{(p^{2}-1)-{{1} \over {p}}} \over {(p-1)}}\right)\left({{p} \over {p^{2}+p+1}}\right)\\[3pt]&=\prod _{p}\left({{(p^{2}-1)-{{1} \over {p}}} \over {(p-1)}}\right)\left({{p} \over {p(p+1+{{1} \over {p}})}}\right)\\[3pt]&=\prod _{p}\left({{(p^{2}-1)-{{1} \over {p}}} \over {p(p-1)}}\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)\\[3pt]&=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{{1} \over {p}} \over {p(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)\end{aligned}}$

$\;\;\;=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{1} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)$

$\;\;\;=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{1} \over {(p-1)}}\right)+\left({{p-1-p^{2}(p-1)} \over {p^{2}(p-1)^{2}}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)$

$\;\;\;=\prod _{p}\left(C_{A}+\left({{(p-1)-p^{2}(p-1)} \over {p^{2}(p-1)^{2}}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)$

$C_{A}$ Artin constant

$\;\;\;=\prod _{p}\left(C_{A}+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)$

Stephens' constant with Landau totient constant and totient constant

$C_{S}=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{1} \over {(p-1)}}\right)+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)$

$C_{S}=\prod _{p}\left(C_{Lt}-\left({{1} \over {p^{2}-p}}\right)-\left({{1} \over {p^{2}-p}}\right)+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)$

$C_{Lt}=$ Landau totient constant

$C_{Lt}=\prod _{p}\left(1+{{1} \over {p(p-1)}}\right)$

$C_{S}=\prod _{p}\left(C_{Lt}-\left({{1} \over {p^{2}-p}}\right)-\left({{1} \over {p^{2}-p}}\right)+C_{t}-\left({{p} \over {p-1}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)$

$C_{t}=$ Totient constant

$C_{t}=\prod _{p}^{}\left(1+{{1} \over {p^{2}(p-1)}}\right)$

See also

• Euler product
• Twin prime constant

References

• (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)

• (Mathworld)http://mathworld.wolfram.com/StephensConstant.html
• (OEIS)https://oeis.org/A078079
• (OEIS)http://oeis.org/A065478
• (OEIS)Sequence in context: A136644 A111963 A206923 * A176982 A079728 A181801
• (Stephens' constant)Stephens, P. J. "Prime Divisor of Second-Order Linear Recurrences, I." J. Number Th. 8, 313–332, 1976.