Oops! Forgot to stipulate: n > 7 . (And don't know why.)
eta(%e^-(2*%pi*sqrt(n))) = (prod(gamma(k/n)^((jacobi(k,n)+1)/4),k,1,n-1))/(sqrt(2)*(a*2^n*%pi^(n+1))^(1/8))
n - 1 jacobi(k, n) + 1 /===\ ---------------- | | 4 k | | gamma (-) | | n - 2 %pi sqrt(n) k = 1 eta(%e ) = ------------------------------ n n + 1 1/8 sqrt(2) (a 2 %pi )
where a is the real root of
3*a^2*\j((%i*sqrt(n)+1)/2)^(1/3)*n+2*a^3 = n^3
2 1/3 %i sqrt(n) + 1 3 3 3 a J (--------------) n + 2 a = n 2
and J is Klein's absolute invariant, IFF sqrt(-n) is a UFD! Apparently you can UFD-test n by solving the eta identity for a and seeing if it satisfies a cubic (with integer coefficients).
There are similar identities for eta(%e^-(%pi*sqrt(n))), eta(%e^-(4*%pi*sqrt(n))), and those log-derivative Lambert type sums. --rwg
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