토션트 상수

토션트 상수(Totient constant) 또는 토션트 합 상수(Totient summatory constant) 는 스티븐스 상수 및 알틴 상수 및 토션트 합 함수(Totient Summatory Function)등과 연관된 수학 상수이다. C t = ∏ p ( 1 + 1 p 2 ( p − 1 ) ) {\displaystyle C_{t}=\prod _{p}^{}\left(1+{{1} \over {p^{2}(p-1)}}\right)} C t = 1.339784153573.... {\displaystyle C_{t}=1.339784153573....} (OEIS의 수열 A065483) C t = ∏ p ( 1 + 1 p 2 ( p − 1 ) ) {\displaystyle C_{t}=\prod _{p}^{}\left(1+{{1} \over {p^{2}(p-1)}}\right)} = ∏ p ( p 2 ( p − 1 ) + 1 p 2 ( p − 1 ) ) {\displaystyle \;\;\;=\prod _{p}^{}\left({{p^{2}(p-1)+1} \over {p^{2}(p-1)}}\right)} = ∏ p ( p 3 − p 2 + 1 p 2 ( p − 1 ) ) {\displaystyle \;\;\;=\prod _{p}^{}\left({{p^{3}-p^{2}+1} \over {p^{2}(p-1)}}\right)} = ∏ p ( − p 2 + 1 p 2 ( p − 1 ) ) + ( p 3 p 2 ( p − 1 ) ) {\displaystyle \;\;\;=\prod _{p}^{}\left({{-p^{2}+1} \over {p^{2}(p-1)}}\right)+\left({{p^{3}} \over {p^{2}(p-1)}}\right)} = ∏ p ( 1 − p 2 p 2 ( p − 1 ) ) + ( p 3 p 2 ( p − 1 ) ) {\displaystyle \;\;\;=\prod _{p}^{}\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)+\left({{p^{3}} \over {p^{2}(p-1)}}\right)} = ∏ p ( 1 − p 2 p 2 ( p − 1 ) ) + ( p p − 1 ) {\displaystyle \;\;\;=\prod _{p}^{}\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)+\left({{p} \over {p-1}}\right)} 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 t = ∏ p ( 1 − p 2 p 2 ( p − 1 ) ) + ( p p − 1 ) {\displaystyle C_{t}=\prod _{p}^{}\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)+\left({{p} \over {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({{1-p^{2}} \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)} = ∏ p ( C A + C t − ( p p − 1 ) ) ( p ( p + 1 + 1 p ) ) {\displaystyle \;\;\;=\prod _{p}\left(C_{A}+C_{t}-\left({{p} \over {p-1}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)} = ∏ p ( ( ( p 2 − 1 ) p ( p − 1 ) ) − ( 1 p − 1 ) + ( 1 − p 2 p 2 ( p − 1 ) ) + ( p p − 1 ) − ( 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-1}}\right)+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)+\left({{p} \over {p-1}}\right)-\left({{p} \over {p-1}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)} = ∏ p ( ( ( p 2 − 1 ) p ( p − 1 ) ) + ( 1 − p 2 p 2 ( p − 1 ) ) + ( p p − 1 ) − ( 1 p − 1 ) − ( p p − 1 ) ) ( p ( p + 1 + 1 p ) ) {\displaystyle \;\;\;=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)+\left({{p} \over {p-1}}\right)-\left({{1} \over {p-1}}\right)-\left({{p} \over {p-1}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)} = ∏ p ( ( ( p 2 − 1 ) p ( p − 1 ) ) + ( 1 − p 2 p 2 ( p − 1 ) ) + 1 − p p − 1 ) ( p ( p + 1 + 1 p ) ) {\displaystyle \;\;\;=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)+1-{{p} \over {p-1}}\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)} = ∏ p ( ( ( p 2 − 1 ) p ( p − 1 ) ) + ( 1 − p 2 p 2 ( p − 1 ) ) + ( p − 1 − p p − 1 ) ) ( p ( p + 1 + 1 p ) ) {\displaystyle \;\;\;=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)+\left({{p-1-p} \over {p-1}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)} = ∏ p ( ( ( p 2 − 1 ) p ( p − 1 ) ) + ( 1 − p 2 p 2 ( p − 1 ) ) − ( 1 p − 1 ) ) ( p ( p + 1 + 1 p ) ) {\displaystyle \;\;\;=\prod _{p}\left(\left({{(p^{2}-1)} \over {p(p-1)}}\right)+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)-\left({{1} \over {p-1}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)} C S = {\displaystyle C_{S}=} 스티븐스 상수 C A = {\displaystyle C_{A}=} 알틴 상수 C t = {\displaystyle C_{t}=} 토션트 상수 스티븐스 상수 란다우 토션트 상수 오일러 피 함수 OEIS OEIS OEIS 매스월드(https://web.archive.org/web/20180315070642/http://mathworld.wolfram.com/TotientSummatoryFunction.html) https://oeis.org/A065483 (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.