It seems that when x = pi sqrt(rational), (sum(n/(%e^(n*x)-1),n,1,inf)+1/(4*x)-1/24)/eta(%e^-x)^4
inf ==== \ n 1 1 > -------- + --- - -- / n x 4 x 24 ==== e - 1 n = 1 -------------------------- 4 -x eta (e )
is algebraic, but disappointingly messy (sqrt cubic surd) for rational = Beegner > 7, as you can judge by the rational = 44 case in http://gosper.org/nuetas.html . Rational = 11 isn't much better.
Understatement! Best I could do with it is Sum[(k/(E^(Sqrt[11]*k*Pi) - 1)), {k, 1, Infinity}]== ((29442373*Gamma[1/22]*Gamma[3/22]*Gamma[5/22]* Gamma[9/22]*Gamma[15/22])/(4224*(-11*2^(1/ 3)*(-88327119*Sqrt[3]*(999716798*Sqrt[11] - 2713716189) + 51490609396526601*Sqrt[11] - 58077417204452290)^(1/ 3) - ((88*2^(2/3)*(16292869731*Sqrt[11] - 38005314908))/(-88327119* Sqrt[3]*(999716798*Sqrt[11] - 2713716189) + 51490609396526601*Sqrt[11] - 58077417204452290)^(1/3)) + 8*(8747*Sqrt[11] - 63877))*(Sqrt[Pi])^7)) - (1/(4*Sqrt[11]* Pi)) + 1/24 {N[%], N[%, 33]} {False, True} FullSimplify has been over a day on it... --rwg