알틴 상수

알틴 상수(Artin Constant) 또는 아르틴 상수는 에밀 아르틴 (Emil Artin)의 이름에서 명명되었다. C A = ∏ n = 1 ∞ ( 1 − 1 P n ( P n − 1 ) ) = 0.3739558136.... ( O E I S A 005596 ) {\displaystyle C_{A}=\prod _{n=1}^{\infty }\left(1-{{1} \over {P_{n}(P_{n}-1)}}\right)=0.3739558136....(OEISA005596)} 이것은 프라임 제타 함수 P ( n ) {\displaystyle P(n)} 와 연결된다. l n C A = − ∑ n = 2 ∞ ( L n − 1 ) P ( n ) n {\displaystyle lnC_{A}=-\sum _{n=2}^{\infty }{{(L_{n}-1)P(n)} \over {n}}} L n {\displaystyle L_{n}} 뤼카 수 , l n {\displaystyle ,ln} 자연로그 C A = ∏ p = p r i m e ( 1 − 1 p ( p − 1 ) ) {\displaystyle C_{A}=\prod _{p=prime}^{}\left(1-{{1} \over {p(p-1)}}\right)} = ∏ p = p r i m e ( 1 − 1 p 2 − p ) {\displaystyle \;\;\;=\prod _{p=prime}^{}\left(1-{{1} \over {p^{2}-p}}\right)} = ∏ p = p r i m e ( p 2 − p − 1 p 2 − p ) {\displaystyle \;\;\;=\prod _{p=prime}^{}\left({{p^{2}-p-1} \over {p^{2}-p}}\right)} = ∏ p = p r i m e ( ( ( p + 1 ) ( p − 1 ) ) − p p 2 − p ) {\displaystyle \;\;\;=\prod _{p=prime}^{}\left({{\left((p+1)(p-1)\right)-p} \over {p^{2}-p}}\right)} = e x p ( ∑ p = p r i m e l n ( ( ( p + 1 ) ( p − 1 ) ) − p ) − l n ( p 2 − p ) ) {\displaystyle \;\;\;=exp\left(\sum _{p=prime}^{}ln(\left((p+1)(p-1)\right)-p)-ln(p^{2}-p)\right)} 알틴상수의 스티븐스 상수 접근표현 C A = ∏ p = p r i m e ( 1 − 1 p ( p − 1 ) ) {\displaystyle C_{A}=\prod _{p=prime}^{}\left(1-{{1} \over {p(p-1)}}\right)} = ∏ p = p r i m e ( p 2 − p − 1 p 2 − p ) {\displaystyle \;\;\;=\prod _{p=prime}^{}\left({{p^{2}-p-1} \over {p^{2}-p}}\right)} = ∏ p = p r i m e ( ( p 2 − 1 ) − p p ( p − 1 ) ) {\displaystyle \;\;\;=\prod _{p=prime}^{}\left({{(p^{2}-1)-p} \over {p(p-1)}}\right)} = ∏ p = p r i m e ( ( p 2 − 1 ) p ( p − 1 ) ) − ( p p ( p − 1 ) ) {\displaystyle \;\;\;=\prod _{p=prime}^{}\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{p} \over {p(p-1)}}\right)} = ∏ p = p r i m e ( ( p 2 − 1 ) p ( p − 1 ) ) − ( 1 ( p − 1 ) ) {\displaystyle \;\;\;=\prod _{p=prime}^{}\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{1} \over {(p-1)}}\right)} C A {\displaystyle C_{A}} 알틴상수 , C S {\displaystyle ,C_{S}} 스티븐스 상수 C A = ∏ p = p r i m e ( ( p 2 − 1 ) p ( p − 1 ) ) − ( 1 ( p − 1 ) ) {\displaystyle C_{A}=\prod _{p=prime}^{}\left({{(p^{2}-1)} \over {p(p-1)}}\right)-\left({{1} \over {(p-1)}}\right)} C S = ∏ p ( ( ( p 2 − 1 ) p ( p − 1 ) ) − ( 1 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^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)} ( 1 ( p − 1 ) ) + x = ( 1 p 2 ( p − 1 ) ) {\displaystyle \left({{1} \over {(p-1)}}\right)+x=\left({{1} \over {p^{2}(p-1)}}\right)} x = ( 1 p 2 ( p − 1 ) ) − ( 1 ( p − 1 ) ) {\displaystyle x=\left({{1} \over {p^{2}(p-1)}}\right)-\left({{1} \over {(p-1)}}\right)} x = ( p − 1 − p 2 ( p − 1 ) p 2 ( p − 1 ) 2 ) {\displaystyle x=\left({{p-1-p^{2}(p-1)} \over {p^{2}(p-1)^{2}}}\right)} x = ( ( p − 1 ) − p 2 ( p − 1 ) p 2 ( p − 1 ) 2 ) {\displaystyle x=\left({{(p-1)-p^{2}(p-1)} \over {p^{2}(p-1)^{2}}}\right)} x = ( ( p − 1 ) ( 1 − p 2 ) p 2 ( p − 1 ) 2 ) {\displaystyle x=\left({{(p-1)\left(1-p^{2}\right)} \over {p^{2}(p-1)^{2}}}\right)} x = ( ( 1 − p 2 ) p 2 ( p − 1 ) ) {\displaystyle x=\left({{\left(1-p^{2}\right)} \over {p^{2}(p-1)}}\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({{\left(1-p^{2}\right)} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)} = ∏ 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)} 2급 알틴상수(Rank 2 Artin constant) C A 2 = ∏ p ( 1 − 1 p 2 ( p − 1 ) ) {\displaystyle C_{A2}=\prod _{p}^{}\left(1-{{1} \over {p^{2}(p-1)}}\right)} 뤼카 수 L-함수 프라임 제타 함수 브룬 상수 란다우 토션트 상수 Artin, E. "Collected Papers" (Ed. S. Lang and J. T. Tate). New York: Springer-Verlag, pp. viii-ix, 1965. (OEIS)https://oeis.org/A065414 (http://mathworld.wolfram.com/TotientSummatoryFunction.html)Niklasch[깨진 링크(과거 내용 찾기)], G. "Some Number-Theoretical Constants." http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml Archived 2006년 6월 12일 - 웨이백 머신. (OEIS)http://oeis.org/A005596 (OEIS)http://oeis.org/A001122 (OEIS)http://oeis.org/A065414 (매스월드) http://mathworld.wolfram.com/ArtinsConstant.html (매스월드) http://mathworld.wolfram.com/FullReptendPrime.html (OEIS)https://oeis.org/A065414 (rank 2 Artin constant)