My former student Kyle Nelson has been playing around with the function f(p) = the product of 1/((sin n pi / p) ^ (n|p)) as n goes from 1 to (p-1)/2 (where p is 1 mod 4 and "(n|p)" is the Legendre symbol) along the lines of the formula that appears at the end of section 13 on page 15 of Barry Mazur's "Princeton Companion to Mathematics" essay "Algebraic Numbers" ( http://www.math.harvard.edu/~mazur/preprints/algebraic.numbers.April.30.pdf) and rediscovering some identities that appear to point to a larger picture that he'd like to know more about. He knows that this stuff isn't new, and has been told that it's related to L-functions for various algebraic number fields, but he'd like some more specific suggestions for what to read. He's been able to show that f(5) = (1 + sqrt(5))/2 f(13) = (3 + sqrt(13))/2 and f(29) = (5 + sqrt(29))/2 so he suspects that when p is of the form d^2 + 4, f(p) equals (d + sqrt(p))/2 (which has the nice continued fraction d + 1/(d + 1/(d + ...). He's also shown that f(41) = 32 + 5 sqrt(41). Anyone know the story that this is part of? Thanks, Jim Propp P.S. This originally came from looking at quotients of gamma functions evaluated at rational numbers, which can be seen by applying the reflection formula to 1 / sin.