* 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).
This first one looks most relevant. Is in on your site? Can you remind me the URL? --Bill
{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}
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