The ternary Goldbach conjecture with primes in thin subsets
type: Item
instance of: scholarly article
title: The ternary Goldbach conjecture with primes in thin subsets
zbMATH Open document ID: 1233.11104
DOI: 10.1216/RMJ-2008-38-2-395
author: C. Bauer
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-04-07T00:00:00Z
review text: The Ternary Goldbach problem, i.e., the problem of representing all odd numbers \(N>5\) as a sum of three primes was established, for all sufficiently large odd \(N\) (say, \(N>N_0\), but with an unknown \(N_0\)), in 1937 by I. M. Vinogradov, applying his trigonometric sums-version of the circle method (of Hardy-Littlewood). Then, we may introduce further constraints on the primes \(p_i\), \(i=1,2,3\), for this equation: \[ N=p_1+p_2+p_3. \tag{\(T\)} \] In fact, the author proves \((T)\) both when the \(p_i\) lie in a constructible thin subset (TS) of the primes and when they are in arithmetic progressions (AP), say \(p_i\equiv b_i\pmod k\), with \(b_i>0\) and \((b_i,k)=1\) (He considers prime moduli \(k\)). Actually, his result for \((T)\) with primes in thin subsets follows, by a standard technique of D. Wolke, from his other AP-result. The story of \((T)\) with the \(p_i\) in TS starts with E. Wirsing, who in 1986 proved the existence of very thin subsets \(S\) of the primes, namely with \[ \left| S\cap [1,x]\right|\ll (x\log x)^{1/3} \] (here the notation \(\ll\) is another gift of I. M. Vinogradov), but \(S\) is not constructible (say, a probabilistic approach prevents us from knowing it). Then, a first explicit construction of a thin \(S\), for \((T)\) with primes in \(S\), was given by A. Balog and J. Friedlander in 1992, using Piatetski-Shapiro primes; this was generalized in 1995 by Jia to the binary Goldbach problem. However, Wolke in 1993 started the standard approach in costructing \(S\), building upon results for arithmetic progressions. We address the interested reader to the paper for a short history about these kind of results. Then, in the present paper the author gives a state-of-the-art result about \((T)\) in AP, namely it can be solved under the additional constraints \(p_i\) in AP of prime moduli \(k\) as above, for ``almost all'\', of them, precisely : \[ \forall F>0 \; \forall \, \text{prime} \, k\leq R:=N^{9/58-\varepsilon} \; \text{except} …
reviewed by: Giovanni Coppola
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 5541226
zbMATH Keywords: ternary Goldbach conjecture
zbMATH Keywords: primes in thin subsets
Wikidata QID: Q123308201
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5541226