Hi Simon, I would bet that your values are related to this: http://www.mathstat.carleton.ca/~williams/papers/pdf/220.pdf Victor (Miller) YOW! So evaluating etas (and thetas and lambdas and ...) is reduced to evaluating L series! Presumably, a solved problem. I had no idea. Simon, your F*(x) := x^(25/24)/eta[x] . Your F*(e^(-2*Pi/5)) result is equivalent to Out[214]= (4 E^(-5 \[Pi]/6) GoldenRatio*Pi^(3/2))/Gamma[1/4]^2 In[215]:= N[%, 22] Out[215]= 0.2000000000000090844043 which really needs to be explained. Your F*(e^(-2*Pi/5)) radical easily denests, and the approximation is Out[246]= 1/10 (5 + 3 Sqrt[5] - 5^(3/4) (1 + Sqrt[5])) -> (E^(-5 Pi/6) Pi^(3/4))/(2^(1/8) Sqrt[2 - 5^(1/4) + 5^(3/4)] Gamma[5/4]) In[247]:= N[%, 33] Out[247]= 0.0887758501511156259651860669196022 -> 0.0887758501511156259651860669653919 which *really* needs to be explained. (The sqrt(trinomial) was reduced from a quadrinomial^(1/8) by Corey's (still unfinished) denester.) --rwg On Wed, Feb 23, 2011 at 12:11 PM, Simon Plouffe <simon.plouffe@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=simon.plouffe%40gmail.com>>wrote:

Hello,

I stumbled on these 2 approximations regarding F(x), the partition function, where p(n) = the number of partitions of n, as usual. aka A000041.

F(x) = sum(p(n)*x^n, n=0..infinity):

Instead I use F*(x) = sum(p(n)*x^(n+1), n=0..infinity):

Then here is the strange thing, for x = exp(-2*Pi/5) then the value is 1/sqrt(5), well almost ; the precision is 13 digits.

For x = exp(-4*Pi/5) the value is 1/2+3/2/sqrt(5)-sqrt(1/2*(1+3/sqrt(5))) the precision is 28 decimal digits. I find this quite surprising. I was sure it was exact, it is NOT. I verified with large values.

Also, apparently these are the only 2 examples I have found within F60 : the Farey fractions up to denominators = 60. Also when x = exp(-Pi/5) = apparently nothing algebraic of a low degree.

caution : do not mistake these values for the standard F(x) which goes 0 for the exponent too, it is not the same.

I added these 2 values in the formulas of A000041 of course.

Does anybody have an idea why these values just pop out like that and apparently no other ???!

Bonne journée. Simon Plouffe