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
inf ==== \ n 1/3
---------------------- = - sqrt(22 / 2 sqrt(11) %pi n ==== %e - 1 n = 1
1/3 1/3 1/3 (2733 sqrt(33) + 624899) + 22 (624899 - 2733 sqrt(33))
1 3 5 9 15 + 240) gamma(--) gamma(--) gamma(--) gamma(--) gamma(--) 22 22 22 22 22
7/2 1 1 /(8448 sqrt(11) %pi ) - -------------- + -- 8 sqrt(11) %pi 24
and closed forms for eta(exp(-2 sqrt(11)pi)) and eta(exp(-2 sqrt(19)pi)). The logderivative sum needs another hour's work.
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