MathOverflow Asked by epic_math on December 15, 2020

Note: This question has been brought here from MSE.

I have been working on various sums involving the zeta function (which come up frequently in my research), and it turned out that many of them had nice closed forms. Today, I was trying to evaluate

$$sum_{rge2}frac{zeta(r)}{r^2}$$

which turned out to be a bit harder than the others. Wolfram Alpha and Mathematica both cannot give the answer, they only give an approximate value of $0.835998$. After half an hour of work, I turned this sum to

$$-int_{0}^{infty}frac{t}{e^t}psi(1-e^{-t})dt-gamma$$

where $gamma$ is the Euler–Mascherni constant and $psi$ is the Digamma function. Now I don’t know what to do further. Does this integral have any closed form? Any help would be appreciated. By the way, some of my ideas to evaluate the integral would be to use some integral or sum representation of the digamma function, and in this case we can interchange the sums and integrals safely.

**Note:** My work is too long to be stated here, so I cannot write it.

**Update:** This integral can be further turned to

$$lim_{ntoinfty}H_n-gamma-sum_{k=1}^{n}mathrm{Li}_2left(frac1kright)$$

where $mathrm{Li}$ is the polylogarithm. I turned this to

$$lim_{ntoinfty}H_n-gamma-sum_{r=1}^{infty}frac{H_{n,r}}{r^2}$$

where $H_{n,r}$ are generalized harmonic numbers.

Get help from others!

Recent Answers

- Lex on Does Google Analytics track 404 page responses as valid page views?
- Jon Church on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP