I think this fully characterizes eta (and (q;q)_oo) near the cyclotomic points: Empirically, 'limit(sqrt(d)*%e^-(%pi^2/(6*d^2*log(q)))*sqrt(-log(q))*eta(%e^(2*%i*%pi*n/d)*q)/(sqrt(2)*sqrt(%pi)),q,1) = exp(%i*%pi*g[n,d]/12) 2 pi - ----------- 2 i pi n 2 ---------- 6 d log(q) d sqrt(d) e sqrt(- log(q)) eta(e q) limit -------------------------------------------------------- q -> 1 sqrt(2 pi) g n, d i pi ------, 12 = e where g[1,d] := 3-d-1/d, g[n,1] := n, and g[n,d] is a sum over the continued fraction expansion of n/d: g[n,d] := if d = 1 then n else (if n = 1 then -d-1/d+3 else (if ?numgcd(n,d) = 1 then block([k : nummod(n,d),f : nummod(d,n)],n/d+g[k,f]-k/d+f/k+1/(f*k)-d/k-1/(d*k)) else "")) 1 (d226) g := if d = 1 then n else (if n = 1 then - d - - + 3 n, d d else (if gcd(n, d) = 1 then block([k : mod(n, d), f : mod(d, n)], n k f 1 d 1 - + g - - + - + --- - - - ---) else "")) d k, f d k f k k d k
2 %pi 2 %i %pi n - ---------- ---------- 726 log(q) 11 limit %e sqrt(- log(q)) eta(%e q) q -> 1
%i %pi k(n) ---------- 2 %pi 132 = sqrt(-----) %e , 11
where k(1),k(2),... = -89, 28, -15, -14, 35, -24, 25, 26, 39, 100 (mod 264). There is probably a nifty formula for these if anyone wants to decrypt it.
(c227) makelist(g[n,11]*11,n,1,10) (d227) [- 89, - 28, - 15, - 14, 35, - 24, 25, 26, 39, 100] I'm no longer shocked Rich didn't see it immediately. And if some Cretaceous Old Legend managed all this by hand, I'd sure like to know how. I think. --rwg