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Sunday, June 27, 2021

Beta function

dirichlet beta

Derivative of Dirichlet Beta function


We prove the following infinite product which turned out to be a modified version of the Dirichlet Beta function derivative. 133557799111113131515=(πeγΓ(34)4)π4
Proof

Take logarithm in both sides ln(3)3+ln(5)5ln(7)7+ln(9)9ln(11)11+ln(13)13...=π4[ln(π)γ4lnΓ(34)]
Then, we want to prove n=1(1)n+1ln(2n1)(2n1)=π4[ln(π)γ4lnΓ(34)]
Consider the Dirichlet beta function β(v)=n=1(1)n+1(2n1)v
Then β(v)=n=1(1)nln(2n1)(2n1)v
Therefore we want to find β(1)=n=1(1)n+1ln(2n1)(2n1)
There are various ways to find the derivative, one of the most straigforward is using the Kummer's Fourier series for the log-gamma function: lnΓ(t)=12ln(πsinπt)+[γ+ln(2π)](12t)+1πn=1lnnnsin(2πnt)
It turned out that if t=14 then sin(πn2)={sin(π2)=1 if n=1sin(π)=0 if n=2sin(3π2)=1 if n=3sin(2π)=0 if n=4}
Then n=1lnnnsin(πn2)=n=1(1)n+1ln(2n1)(2n1) Therefore, n=1(1)n+1ln(2n1)(2n1)=π[lnΓ(14)12ln(πsinπ4)[γ+ln(2π)](14)]=π[ln(π)12ln(2)lnΓ(34)12ln(π)14ln(2)14γln(2π)4]=π4[ln(π)γ4Γ(34)]
Therefore 133557799111113131515=(πeγΓ(34)4)π4

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