If we assume the Generalized Riemann Hypothesis, the argument used for the major arcs can be extended to the minor arcs. This was done by Hardy and Littlewood in 1923. In 1937 Vinogradov gave an unconditional upper bound for |S(\alpha)|. His argument began with a simple sieve identity, the resulting terms were then rearranged in a complicated way to obtain some cancellation. In 1977 R. C. Vaughan found a much simpler argument, based on what later became known as Vaughan's identity. He proved that if |\alpha-\frac{a}{q}|<\frac{1}{q^2}, then
