* rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> [Jul 02. 2009 09:56]:
[...]
Eye mercy: http://gosper.org/newetas.html
Working toward sqrt(163)pi.
These are purely empirical, unproven results. Incredibly, I'm doing the numerics in Macsyma instead of Mma due to bizarre precision bugs. And bizarreness in general. Floor[<numeric infinite series>] gave no integer. N[%] does, but then N[%] again makes a short float!
But Mma's algebraic number stuff is pretty impressive. Still doesn't denest, tho.
I shouldn't jinx myself, but I think I can do exp(pi sqrt(n/d)) for n and d "within reason". If 163 is beyond reason, wait 'til next year. --rwg
The following refs might be helpful (wanted to work on that myself, don't have time). {Jinhee Yi: {Theta-function identities and the explicit formulas for theta-function and their applications}, Journal of Mathematical Analysis and Applications, vol.292, no.2, pp.381-400, \bdate{15-April-2004}. \jjfile{yi-theta-func-identities.pdf} % Seems to recycle vasuki-note-on-PQ-modeq.pdf {K.\ R.\ Vasuki, T.\ G.\ Sreeramamurthy: {A Note on $P$-$Q$ Modular Equations}, Tamsui Oxford Journal of Mathematical Sciences, vol.21, no.2, pp.109-120, \bdate{2005}. URL: \url{http://www.mcs.au.edu.tw/vol-21-2.htm}.} \jjfile{vasuki-note-on-PQ-modeq.pdf} % Much of this seems to be recycled in yi-theta-func-identities.pdf {M.\ S.\ Mahadeva, H.\ S.\ Madhusudhan : {Some explicit values for ratios of theta-functions}, General Mathematics, vol.13, no.2, pp.105-116, \bdate{2005}. URL: \url{http://www.emis.de/journals/GM/vol13nr2/cuprins132.html}.} \jjfile{mahadeva-some-explicit-theta-values.pdf}