■もうひとつのバーゼル問題(その21)

[0]∫(0,1)logx/(x−1)dx=ζ(2)=π^2/6

[1]∫(0,1)(x−1)/logxdx=log2

[0]∫(0,1)(logx)^2/(x−1)^2dx=2ζ(2)=π^2/3

[1]∫(0,1)(x−1)^2/(logx)^2dx=log(27/16)

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 Ir=∫(0,1)(logx/(x−1))^rdx

 Jr=∫(0,1)((x−1)/logx)^rdx

とおくと,(

 I1=ζ(2)=π^2/6

 I2=2ζ(2)=π^2/3

 I3=3ζ(3)+3ζ(2)

 I4=8ζ(4)+12ζ(3)+4ζ(2)

 J1=log2

 J2=log(27/16)

 J3=1/2・log(2^44/3^27)

 J4=1/6・log(3^165^125/2^544)

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