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

 

[math]\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[/math]

 

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.lbl.gov/~dhbailey/dhbpapers/math-future.pdf

 

Thanks,

Daniel

Edited September 28, 20169 yr by renerpho

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.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf.

Edited September 29, 20169 yr by renerpho