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A complete Vinogradov 3primes theorem under the Riemann hypothesis
 ERA Am. Math. Soc
, 1997
"... Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1. ..."
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Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1.
Let SUMS OF THREE OR MORE PRIMES
"... Abstract. It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error � p≤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. ..."
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Abstract. It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error � p≤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 ≥ 4, and, in a mean square sense, for k ≥ 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 “QuasiRiemann Hypothesis”. We incidentally give a slight sharpening to a wellknown exponential sum estimate of VinogradovVaughan. r(n)=rk(n)= 1.