I just got SUM(N/(%E^(2*SQRT(11)*%PI*N)-1),N,1,INF) = -SQRT(22^(1/3)*(2733*SQRT(33)+624899)^(1/3)+22^(1/3)*(624899-2733*SQRT(33))^(1/3)+240)*GAMMA(1/22)*GAMMA(3/22)*GAMMA(5/22)*GAMMA(9/22)*GAMMA(15/22)/(8448*SQRT(11)*%PI^(7/2))-1/(8*SQRT(11)*%PI)+1/24
Joerg Arndt has just sent me a 2004 paper (THANKS!) by J. Yi actually deriving closed forms for theta[3](0,q) for q = +-exp(-n pi) for n as large as 12, in terms of <algebraic>*pi^(1/4)/Gamma(3/4). This is my first inkling that this eta stuff I've been bombarding you with isn't new. (Eavesdropper Bruce Berndt submitted the paper. Bruce, can you tell me what's known about Theta/Eta special values? (Not quotients.)) I got eta(exp(-pi Sqrt(43))) as a nice algebraic times a *horrible* pile of Gamma(k/86)^(n/10), for nearly all 0<k<43. The logs of all those Gammas resist numerical relation-finding. Those tenth roots are news, too. The Lambert series can't be much better: different algebraic times same Gammas to different powers. --rwg
Eye mercy: http://gosper.org/newetas.html
Does anybody really want to see eta(exp(-pi Sqrt(43)))? If you have Mma, DedekindEta[I*Sqrt[43]] == ((-(80/(1 + 63*Sqrt[129])^(1/3)) + (1 + 63*Sqrt[129])^(1/3))^(1/8)* Gamma[7/43]^(1/5)*Gamma[11/43]^(9/10)*Gamma[15/43]^(7/10)* (Gamma[4/43]*Gamma[6/43]*Gamma[10/43]*Gamma[14/43]* Gamma[16/43])^ (2/5)*(Gamma[3/43]*Gamma[19/43])^(3/5)* ((Gamma[1/86]*Gamma[9/86]*Gamma[13/86]*Gamma[17/86]* Gamma[21/86]* Gamma[25/86]*Gamma[41/86])/(Gamma[2/43]*Gamma[8/43]* Gamma[12/43]* Gamma[18/43]*Gamma[20/43]))^(1/10)* ((Gamma[1/43]*Gamma[5/43]*Gamma[9/43]*Gamma[13/43]* Gamma[17/43]* Gamma[21/43])/(Gamma[7/86]*Gamma[15/86]*Gamma[23/86]* Gamma[31/86]*Gamma[39/86]))^(3/10))/(2*86^(7/40)* Pi^(17/20)* Sqrt[Gamma[11/86]]*(Gamma[3/86]*Gamma[19/86])^(7/10)* Gamma[27/86]^(3/5)*Gamma[35/86]^(1/5)* (Gamma[5/86]*Gamma[29/86]*Gamma[33/86]*Gamma[37/86])^(2/5))
Working toward sqrt(163)pi.
It's threatening to be hideous. --rwg
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
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