renerpho Posted September 28, 2016 Share Posted September 28, 2016 (edited) 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, 2016 by renerpho Link to comment Share on other sites More sharing options...
renerpho Posted September 29, 2016 Author Share Posted September 29, 2016 (edited) 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, 2016 by renerpho Link to comment Share on other sites More sharing options...
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