On Vinogradov's constant in Goldbach's ternary problem
The author shows that under the assumption of the Generalized Riemann Hypothesis every odd integer greater than \(10^{20}\) can be written as a sum of three primes. The proof requires a careful numerical study of the exponential sum \[ S(\alpha) =\sum_{n>1} \Lambda(n) \exp(-2\pi i\alpha n-n/N). \] The best value hitherto (under GRH) was \(10^{32}\) [\textit{B. Lucke}, Zur Hardy-Littlewoodschen behandlung des Goldbachschen Problems. Göttingen: Math.-naturwiss. Diss. (1926; JFM 52.0167.02)]. The best unconditional constant is about \(10^{43.000}\) [\textit{J. Chen} and \textit{T. Wang}, Acta Math. Sin. 32, 702-718 (1989; Zbl 0695.10041)].
scientific article; zbMATH DE number 1048759