스티븐스 상수

스티븐스 상수(Stephens' Constant)에 대한 설명이다. ∏ p ( 1 − p p 3 − 1 ) = 0.57595996889294543964316337549249669... {\displaystyle \prod _{p}\left(1-{{p} \over {p^{3}-1}}\right)=0.57595996889294543964316337549249669...} (OEIS의 수열 A065478) 스티븐스상수는 소수 및 오일러의 곱셈 공식과 관련하여 소수 분포 밀도에대한 수학 상수이다. 스티븐스(Stephens, P. J.)는 일반화된 리만 가설을 가정하고서, 소수에 대한 집합의 밀도를 표현해 보였다. C S = ∏ p ( 1 − p p 3 − 1 ) {\displaystyle C_{S}=\prod _{p}\left(1-{{p} \over {p^{3}-1}}\right)} 과정: = ∏ 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}} 는 알틴 상수 = ∏ 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 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 L t = {\displaystyle C_{Lt}=} 란다우 토션트 상수 C L t = ∏ p ( 1 + 1 p ( p − 1 ) ) {\displaystyle C_{Lt}=\prod _{p}\left(1+{{1} \over {p(p-1)}}\right)} C t = {\displaystyle C_{t}=} 토션트 상수 C t = ∏ p ( 1 + 1 p 2 ( p − 1 ) ) {\displaystyle C_{t}=\prod _{p}^{}\left(1+{{1} \over {p^{2}(p-1)}}\right)} 토션트 상수 쌍둥이 소수 상수 (Stephen 1976)Stephens, P. J. "Prime Divisor of Second-Order Linear Recurrences, I." J. Number Th. 8, 313-332, 1976. Mathematical Constants (공)저: 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) (매스월드)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.