{{위키데이터 속성 추적}}
'''토션트 상수'''(Totient constant) 또는 '''토션트 합 상수'''(Totient summatory constant)<ref>https://oeis.org/A065483</ref><ref>(http://mathworld.wolfram.com/TotientSummatoryFunction.html)Niklasch{{깨진 링크|url=http://mathworld.wolfram.com/TotientSummatoryFunction.html)Niklasch }}, G. "Some Number-Theoretical Constants." http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml {{웨이백|url=http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml |date=20060612212955 }}.</ref>
는 [[스티븐스 상수]] 및 [[알틴 상수]] 및 [[토션트 합 함수]](Totient Summatory Function)등과 연관된 [[수학 상수]]이다.

:<math>C_t=\prod_{p}^{} \left( 1+{{1}\over{p^2(p-1)}} \right)</math>

<math>C_t= 1.33978 41535 73....</math>  {{OEIS|A065483}}

==토션트 상수 변형==

:<math>C_t=\prod_{p}^{} \left( 1+{{1}\over{p^2(p-1)}} \right)</math>
:<math>\;\;\;=\prod_{p}^{} \left( {{p^2(p-1)+1}\over{p^2(p-1)}} \right)</math>
:<math>\;\;\;=\prod_{p}^{} \left( {{p^3-p^2+1}\over{p^2(p-1)}} \right)</math>
:<math>\;\;\;=\prod_{p}^{} \left( {{-p^2+1}\over{p^2(p-1)}} \right) + \left( {{p^3}\over{p^2(p-1)}} \right)</math>
:<math>\;\;\;=\prod_{p}^{} \left( {{1-p^2}\over{p^2(p-1)}} \right) + \left( {{p^3}\over{p^2(p-1)}} \right)</math>
:<math>\;\;\;=\prod_{p}^{} \left( {{1-p^2}\over{p^2(p-1)}} \right) + \left( {{p}\over{p-1}} \right)</math>

==스티븐스 상수에서  알틴 상수와 토션트 상수==
:<math> C_A= \prod_{p=prime}^{} \left({{(p^2 -1)}\over{p(p -1)}}\right)- \left({{1}\over{p -1}}\right)</math>
:<math>C_t=\prod_{p}^{} \left( {{1-p^2}\over{p^2(p-1)}} \right) + \left( {{p}\over{p-1}} \right)</math>
:<math>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)</math>

:<math>\;\;\;= \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)</math>
:<math>\;\;\;= \prod_{p} \left( C_A + C_t - \left( {{p}\over{p-1}} \right)    \right)
\left({{p}\over{(p+1+{{1}\over{p}})}} \right)</math>

:<math>\;\;\;= \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)</math>
:<math>\;\;\;= \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)</math>
:<math>\;\;\;= \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)</math>
:<math>\;\;\;= \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)</math>
:<math>\;\;\;= \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)</math>



:<math> C_S=</math>[[스티븐스 상수]]
:<math> C_A=</math>[[알틴 상수]]
:<math> C_t=</math> 토션트 상수

== 같이 보기 ==
* [[스티븐스 상수]]
* [[란다우 토션트 상수]]
* [[오일러 피 함수]]

==참고==
* [https://oeis.org/A065483 OEIS]
* [http://oeis.org/A065484 OEIS]
* [http://oeis.org/A118262 OEIS]
*매스월드(https://web.archive.org/web/20180315070642/http://mathworld.wolfram.com/TotientSummatoryFunction.html)

== 각주 ==
{{각주}}

[[분류:수학 상수]]