란다우 토션트 상수 문서 원본 보기

{{위키데이터 속성 추적}}
'''란다우 토션트 상수'''(Landau's totient constant)<ref>(OEIS A082695), mu(k) is the Möbius function, zeta(z) is the Riemann zeta function, and p_k is the kth prime (Landau 1900; Halberstam and Richert 1974, pp. 110-111; DeKoninck and Ivić 1980, pp. 1-3; Finch 2003, p. 116; Havil 2003, p. 115; Dickson 2005)</ref>

:<math>C_{Lt} = \prod_{p} \left(1 + {{1}\over{p(p-1)} } \right) = {{315}\over{2\pi^4} }\zeta(3) = 1.943596436820759205057070... </math> {{OEIS|A082695}}

:<math>C_{Lt} = {{315}\over{2\pi^4} }\zeta(3)  = {{1}\over{{\zeta(2) \zeta(3)}\over{\zeta(6)}} }= {{\zeta(2) \zeta(3)}\over{\zeta(6)}} = \sum_{s=1}^{\infty} {{(\mu(s))^2}\over{s \phi(s)}} \;\;\; </math><ref>Decimal expansion of 315/(2*Pi^4 ) - http://oeis.org/A157292</ref><ref>(Decimal expansion of zeta(6)/(zeta(2)*zeta(3)))http://oeis.org/A068468</ref><ref>(Decimal expansion of zeta(2)*zeta(3)/zeta(6))http://oeis.org/A082695</ref>

:<math>\zeta(p)</math>[[리만 제타 함수]]<math>, \;\; \mu(s)  </math> [[뫼비우스 함수]]<math> , \;\;\phi(s) </math> [[오일러 피 함수]]


==알틴 상수와 란다우 토션트 상수==
:<math>C_{Lt} = </math> 란다우 토션트 상수
:<math>C_{Lt} = \prod_{p} \left(1 + {{1}\over{p(p-1)} } \right)</math>
:<math>\;\;\; = \prod_{p} \left({{p^2-p+1}\over{p^2-p} } \right)</math>
:<math>\;\;\; = \prod_{p} \left({{p^2-p}\over{p^2-p} } \right) + \left({{1}\over{p^2-p} } \right)</math>

:<math>C_{A} = </math> [[알틴 상수]]
:<math> C_{A}= \prod_{p}^{}  \left( 1-{{1}\over{p(p -1)}}\right)</math>
:<math> \;\;\;= \prod_{p}^{} \left({{p^2 -p-1}\over{p^2 -p}}\right)</math>
:<math>\;\;\; = \prod_{p} \left({{p^2-p}\over{p^2-p} } \right) - \left({{1}\over{p^2-p} } \right)</math>

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

:<math>C_{A} =  C_{Lt} -  \left({{1}\over{p^2-p} } \right) -  \left({{1}\over{p^2-p} } \right)</math>

== 같이 보기 ==
* [[스티븐스 상수]]
* [[토션트 상수]]

== 각주 ==
{{각주}}
* [https://oeis.org/A014197 OEIS]
* [https://oeis.org/A059956 OEIS]
* [https://web.archive.org/web/20180315070642/http://mathworld.wolfram.com/TotientSummatoryFunction.html 매스월드]
* [http://mathworld.wolfram.com/TotientValenceFunction.html 매스월드]

[[분류:수학 상수]]