show that
∞∑n=1ζ2n4=ζ2(3)−13ζ(6)
where
ζm=n∑k=11km,ζ(m)=∞∑k=11km
is true?
because This result is my frend tell me.
This problem have someone research it?Thank you
my some idea:
ζ3(3)=(∞∑n=01(n+1)3)2=∞∑n=0n∑k=01(k+1)3(n−k+1)3
and use
1(k+1)(n−k+1)=1n+2(1k+1+1n−k+1)
and (a+b)3=a3+3a2b+3ab2+b3
and
∞∑n=1Hn(n+1)5=12(5ζ(6)−2ζ(2)ζ(4)−ζ2(3))
But is very ugly, someone have other nice methods? Thank you .
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