로그 탄젠트 적분(log tangent integral)

\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{\pi}{2}\ln{\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\)

\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{d}{ds}\Gamma(s)L(s)|_{s=1}\)

\(F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}\)

\(\Gamma(s)F(s)=\int_0^{\infty}(\sum_{n=1}^{\infty}f(n)e^{-nt})t^{s-1}\,dt\)\(z=e^{-t}\) 로 치환하면,

\(\Gamma(s)F(s)=\int_0^{1}(\sum_{n=1}^{\infty}f(n)z^n)(\log\frac{1}{z})^{s-1}\,\frac{dz}{z}\)

\(f(n+q)=f(n)\) 을 만족하면 (가령 디리클레 캐릭터의 경우)

\(\sum_{n=1}^{\infty}f(n)z^n=\frac{p(z)}{1-z^q}\)

여기서 \(p(z)=\sum_{n=1}^{q-1}f(n)z^n\)

이를 이용하면,

\(\Gamma(s)F(s)=\int_0^{1}\frac{p(z)(\log\frac{1}{z})^{s-1}}{1-z^q}\,\frac{dz}{z}\) 를 얻는다.

\(\frac{d}{ds}\Gamma(s)F(s)=\int_0^{1}\frac{p(z)(\log\frac{1}{z})^{s-1}}{1-z^q}\log \log\frac{1}{z} \,\frac{dz}{z}\)

[1]Victor H. Moll

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