Jump to content


Photo
- - - - -

Curious unproven integral formula


  • Please log in to reply
1 reply to this topic

#1 renerpho

renerpho

    Quark

  • Members
  • 20 posts
  • LocationGermany

Posted 28 September 2016 - 02:13 AM

Has there ever been a formal proof of the following formula?

 

\sum_{n=0}^\infty {1\over (7n+1)^2}+{1\over (7n+2)^2}-{1\over (7n+3)^2}+{1\over (7n+4)^2}-{1\over (7n+5)^2}-{1\over (7n+6)^2} \stackrel{?}{=}{24\over 7\sqrt{7}}\int_{\pi/3}^{\pi/2} \! \log {|{\tan{t}+\sqrt{7}\over \tan{t}-\sqrt{7}}|} \mathrm{d}t \approx 1.15192

 

The most recent result I can find is from Bailey&Borwein (2005), who have shown that the identity holds to at least 20,000 decimal places. Yet, no proof that it is exact has been known at that time.

The Bailey&Borwein (2005) paper can be found here: http://crd-legacy.lb...math-future.pdf

 

Thanks,

Daniel


Edited by renerpho, 28 September 2016 - 02:25 AM.

  • 0

#2 renerpho

renerpho

    Quark

  • Members
  • 20 posts
  • LocationGermany

Posted 29 September 2016 - 09:26 PM

Remark: Bailey, Borwein et al. (2006) give a nice heuristic argument why this formula might be true, which is related to double Euler sums and 4-dimensional geometry, as well as Quantum Physics. See p.11 of http://crd-legacy.lb...tenproblems.pdf.
ionIZOx.png


Edited by renerpho, 29 September 2016 - 09:27 PM.

  • 0




0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users