×

On Vinogradov’s constant in Goldbach’s ternary problem. (English) Zbl 0876.11047

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 \(10^{32}\) [B. Lucke, Zur Hardy-Littlewoodschen behandlung des Goldbachschen Problems. Göttingen: Math.-naturwiss. Diss. (1926; JFM 52.0167.02)]. The best unconditional constant is about \(10^{43.000}\) [J. Chen and T. Wang, Acta Math. Sin. 32, 702-718 (1989; Zbl 0695.10041)].

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Karatsuba, A. A., Basic Analytic Number Theory (1991), Springer-Verlag: Springer-Verlag New York/Berlin
[2] Linnik, U. V., The new proof of Goldbach-Vinogradov’s theorem, Mat. Sb., 19, 3-8 (1946) · Zbl 0063.03589
[3] Chen, J.-R.; Wang, T.-Z., On odd Goldbach problem under general Riemann hypothesis, Sci. China, 36, 682-691 (1993) · Zbl 0779.11044
[4] Schoenfeld, L., Sharper bounds for the Chebyshev functions \(θxψx\), Math. Comput., 30, 337-360 (1976) · Zbl 0326.10037
[5] B. Lucke, 1926, Zur Hardy-Littlewoodschen Behandlung des Goldbachschen Problems, Univ. of Göttingen; B. Lucke, 1926, Zur Hardy-Littlewoodschen Behandlung des Goldbachschen Problems, Univ. of Göttingen · JFM 52.0167.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.