Hello mr. Gosper, now that is interesting, you have the explicit formula for sum(1/n/(exp(Pi*n*k)-1),n=1..infinity) when k=1,2,3,4,5,6. Here with PSLQ or LLL I could get 1,2 and 4 only separatly in terms of log(Pi), Pi, log(2) and log(GAMMA(1/4)), I did not know that one could get those explicit algebrico-log-gamma expressions . What is surprising is the <cancel out> of the algebraic expression when k = 1/5, 2/5 and 4/5 and also the approximations when k = 2/13, 2/7 or 2/163, I tried to find other fractions for k and found only that explicit 1/5, 2/5 and 4/5. Nevertheles, I have found a simpler formulation of the formula for pi ; it appears on my home page at http://www.plouffe.fr/ I should switch to macsyma! best regards, Simon Plouffe