Let f(x) and g(x) be rational functions of x with integer coefficients, p an odd prime, e(x) = e(2πix/p), e(u/v) = e(w), where u = vw (mod p)† and χ(x) is a character mod p. It is well known that very abstruse methods are required to find an estimate for sums such as

where in the summation, ‡ the values of x are omitted for which eitherf(x) or g(x) is not defined.