How to prove
∞∑n=1H2nH(2)n(2n+1)2=43ln32ζ(2)−72ln22ζ(3)−2116ζ(2)ζ(3)+71364ζ(5)−415ln52−8ln2Li4(12)−8Li5(12)
where H(q)n=∑nk=11nq is the harmonic number, Lir(x)=∑∞n=1xnnr is the polylogarithm function and ζ is the Riemann zeta function.
This problem is proposed by Cornel with no solution submitted.
My trial
By applying integration by parts we have
∫10x2n(Li2(x)−ζ(2)) dx=−H2n(2n+1)2−1(2n+1)3
now multiply both sides by H(2)n then sum both sides from n=1 to ∞ we get
∫10(Li2(x)−ζ(2))∞∑n=1H(2)nx2n dx=−∞∑n=1H2nH(2)n(2n+1)2−∞∑n=1H(2)n(2n+1)3
∫10(Li2(x)−ζ(2))Li2(x2)1−x2 dx=−∞∑n=1H2nH(2)n(2n+1)2−∞∑n=1H(2)n(2n+1)3
I managed to find the blue sum using Abel's summation. As for the integral, I tried integration by parts but still resistant.
QUESTION
Any idea how to crack the integral or a different approach to find the target sum?
Thanks.
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