For odd d, 'limit(theta[3](0,%e^(2*%i*%pi*n/d)*r)*sqrt(-log(r)),r,1) = sqrt(%pi/d)*%e^(%i*%pi*(-2*g(4*n/d)+5*g(2*n/d)-2*g(n/d))/(12*d))
2 i pi n -------- d limit theta (0, e r) sqrt(- log(r)) = r -> 1 3
4 n 2 n i pi (- 2 g(---) + 5 g(---) - 2 g(n/d)) d d --------------------------------------- pi 12 d sqrt(--) e d
Where, as with the eta limits, g(r) := if integerp(r) then r else (floor(r),denom(r)*(%%+3)-(g(1/(r-%%))+1/denom(r))/(r-%%))
g(r) := if integerp(r) then r else (floor(r),
1 1 g(------) + -------- r - %% denom(r) denom(r) (%% + 3) - --------------------), r - %% a n(ot obviously) integer-valued function.
The exponent expression -2*g(4*n/d)+5*g(2*n/d)-2*g(n/d)
4 n 2 n n - 2 g(---) + 5 g(---) - 2 g(-) d d d
appears to be 0 mod 6, period n
2n
for fixed n, period d for fixed d, but otherwise no simpler than g. Even d looks not much harder. (Betcha you reparsed!) --rwg PS, how can a natural boundary grow only like 1/sqrt(-log q)? The other thetas are *much* wilder.
This question is braindead thrice over. Theta_4 goes "wildly" to 0 similarly to the oddly even d case of theta_3. Theta_2 (odd d) is similar to theta_3: theta[2](0,%e^(2*%i*%pi*n/d)*r)*sqrt(-log(r)) = sqrt(%pi/d)*%e^(%i*%pi*(2*g(4*n/d)-g(2*n/d))/(12*d)) 2 i n pi -------- d theta (0, e r) sqrt(- log(r)) = 2 4 n 2 n i (2 g(---) - g(---)) pi d d ------------------------ 12 d pi e sqrt(--) d The poles are at irrational points. Duh. PS, almost exactly a week after buying it, Greg Whitehead has become the first solver of the 82% Arnold Dozenegger other than Emma Cohen, who actually took more than a week to rediscover pretty nearly her own design, despite her abundant free time as a Cal Tech student.