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I am trying make interesting integrals involving the fractional part function and special functions. I wondered if it is possible to deduce a series representation (in the atempt to get a closed-form that maybe there no exists) for $$\int_0^{1/3} \left\{ \frac{1}{x} \right\}W_0(x)dx,$$ where $ \left\{ x \right\}=x-\lfloor x\rfloor$ denotes the fractional part function and $W_0(x)$ the Lambert W-Function, see for example the related MathWorld's article Lambert W-Function or the corresponding Wikipedia.

Claim. One has that

$$\int_0^{1/3} \left\{ \frac{1}{x} \right\}W_0(x)dx=\sum_{n=1}^\infty\frac{(-1)^{n-1}n^{n-2}}{(n+1)!}\left(3^{-n}-n\left(\zeta(n+1)-1-2^{-(n+1)}-3^{-(n+1)}\right)\right),$$ where $\zeta(s)$ denotes the Riemann zeta function.

Question. I think that previous Claim is right. Am I right? Are feasible more simplifications (closed-forms for some of the terms of the series of RHS, or a better way to write the resulting series) for my deduction? Many thanks.

I would like to know if there are more potentially interesting combinations of definite integrals involving the fractional part function and the Lambert W-function (for a different integrand if it is required). Thus if you want answer next optional question adding a paragraph in your answer or well in comments.

Question (Optional). Can you propose a more interesting (with a nice closed-form or with a calculation more interesting than mine) definite integral involving the fractional part function and a Lambert W-function (the interval of integration can be different than mine)? Many thanks.

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  • $\begingroup$ How did you get this series expression and have you checked it numerically? $\endgroup$ – Yuriy S Oct 17 '18 at 22:22
  • $\begingroup$ First many thanks for your friendly behaviour with all users in this site MSE. Secondly I don't remember how was deduced my claim, but I think that I've combined the series expansion of the W-Lambert function and integration of fractional parts to deduce my result, that I don't checked numerically. In Question I am asking about if my deduction is right or as companion interesting results or simplifications related to mine @YuriyS Feel free to add your contribution so that other users can value it, I am going to delete my account in this site, but your contributions are always valuable. $\endgroup$ – user243301 Oct 19 '18 at 10:02

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