Sums of Three or More Primes

It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error $\sum _{p \le x} \log p - x$ in the Prime Number Theorem, such bounds being within a factor of $(\log x)^{2}$ of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided ``Riemann Hypothesis'' is replaced by ``Generalized Riemann Hypothesis'', results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of $k$ primes for $k \ge 4$, and, in a mean square sense, for $k \ge 3$. We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a ``Quasi-Riemann Hypothesis''. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.