where [[zeta].sub.d] is a primitive d-th

root of unity. We denote by [[lambda].sub.d] this matrix generating [G.sub.P].

So if [lambda] = exp(2k[pi]i/n) is a primitive nth

root of unity and q | n, then [[lambda].sup.q] = exp(2k[pi] i/(n/q)) is a primitive (n/q)th

root of unity.

(1) There is no single [theta]-function which works for all [zeta] i.e., for every [theta]-function [theta](q) there is some

root of unity [zeta] for which difference f(q) - [theta](q) is unbounded as q [right arrow] [zeta] radially.

If d [greater than or equal to] 3 does not divide k, then c(n, k; w) = 0, where w is a primitive d-th

root of unity.

Let K be the field of complex numbers C and q be a primitive Nth

root of unity, where N [greater than or equal to] 2.

(i) F vanishes at some primitive d-th

root of unity;

Throughout this work we assume that if p [member of] I then p is not a

root of unity, so that we always have [I.sub.[lambda]] = [I.sup.+.sub.[lambda]] [??] [I.sup.-.sub.[lambda]].

This paper studies the cyclotomic polynomial [[PHI].sub.n](x), which is defined as the minimal polynomial over Q for any primitive nth

root of unity [zeta] in C.

If e is invertible on Z, Z/l (1) denotes the sheaf of l-th

root of unity and for any integer i, we denote Z/l (i) = [(Z/l (1)).sup.[cross product]i].

Since every root of [mu](x) mod P coincides with [theta] mod P for some choice of primitive nth

root of unity [[xi].sub.n], we see, by Lemma 4.3, that [mu](x) and g(x) have no common roots over [bar.Fp].

where [w.sub.d] is a primitive d-th

root of unity. In other words, the evaluations of the polynomial X(q) at appropriate roots of unity carry all the numerical information about the C-orbit structure.

Indeed, if a primitive p-th

root of unity [[zeta].sub.p] is in F([square root of -2[xi]), then Q([[zeta].sub.p]) [subset] F([square root of -2[xi]), so 2g = [F([[zeta].sub.p]): Q] [greater than or equal to] [Q([[zeta].sub.p]) : Q] = p - 1.