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I've been going through "103 (almost) impossible integrals, sums and series" which is a book containing lots of interesting integrals. I saw a very intriguing definite integral that the book stated has a relation with Riemann's zeta function in the following manner: $$ \int_{0}^{1} \frac{\ln^2(x+1)}{x} dx = \frac{1}{4}\zeta(3) $$ I thought this connection's meaning would become clear when I prove it but now that I've proved the relation I still feel uncomfortable with it. It's just so out of place. So my question is, what is the intuition behind this equation? Why would zeta function have anything to do with this particular function's integral?

tripleee
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hooman hedayti
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  • Because you can eventually get zeta-like sums, as in some answers here? I'm ... not entirely sure what kind of answer you expect. – PrincessEev Aug 16 '22 at 20:45
  • Actually, that identity is trivial to solve, after looking at the definition of the Zeta function, and placing a change of variables. – Brethlosze Aug 16 '22 at 20:46
  • @princessEev i know how this would mathematically be connected my problem is it's meaning. I can't see a reason why these should be connected but they are. Maybe i shouldn't search for deeper meanings? Idk – hooman hedayti Aug 16 '22 at 20:50
  • @brethlosze i used the gamma function representation of zeta function and then turned gamma function to beta function and wrote beta function's integral form and then did some change of variables to match the boundries of integration. I wonder how you did it because mine didn't meant anything. – hooman hedayti Aug 16 '22 at 20:53
  • How would you find the intuition between 2 steps when evaluating the integral? – Тyma Gaidash Aug 16 '22 at 22:11
  • @tymaGaidash i don't understand the relation of gamma and zeta functions that well. I know how it's derived but i don't know what's the reason behind the connection. Is understanding this a good start ? – hooman hedayti Aug 17 '22 at 07:32

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