Binary Goldbach problem with restrictions on the summands
type: Item
instance of: scholarly article
title: Binary Goldbach problem with restrictions on the summands
zbMATH Open document ID: 0824.11063
DOI: 10.1007/BF01876626
author: Gunter Dufner
published in: Periodica Mathematica Hungarica
publication date: 1995-06-28T00:00:00Z
review text: The author studies the number of Goldbach representations of even numbers in which one summand is very small. Under GRH he obtains that for any \(C_ 1>0\), \(C> 5C_ 1+ 44\) and \(H= (\log x)^ C\), for all even \(N\leq x\) with at most \(\ll x(\log x)^{-C_ 1}\) exceptions, we have \(|\{ (p,p_ 1)\): \(p+p_ 1= N\) and \(p_ 1\leq H\}| \sim \sigma(N) H/ (\log x\log H)\), where \(\sigma\) denotes the expected singular series. Moreover, there exists a thin set of primes \(\mathbb{P}_ 1\) (that is, \(| \mathbb{P}_ 1\cap [1, x]| \ll (\log x)^{C+ 1})\) such that all even \(N\leq x\) with at most \(\ll x(\log x)^{- C_ 1}\) exceptions, have \(\gg \sigma (N) (\log x)^{C -2}\) Goldbach representations \(N= p+ p_ 1\), with \(p_ 1\in \mathbb{P}_ 1\). \textit{G. Dufner} and \textit{D. Wolke} [Montash. Math. 118, 75-82 (1994; Zbl 0808.11060)] proved that, unconditionally, there exists a set \(\mathbb{P}_ 2\) with \(| \mathbb{P}_ 2\cap [1, x]|\ll (\log x)^ 2\) such that the equation \(N= p+ p_ 1\), with \(p_ 1\in \mathbb{P}_ 2\) has at least one solution for almost all even \(N\leq x\) as above. The author also gives several similar results on ternary Goldbach representations of odd integers.
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11B13
zbMATH DE Number: 753671
zbMATH Keywords: binary Goldbach problem
zbMATH Keywords: conditional results
zbMATH Keywords: number of Goldbach representations of even numbers
zbMATH Keywords: ternary Goldbach representations
reviewed by: Alessandro Zaccagnini
MaRDI profile type: Publication
cites work: Proof of a heuristic principle in additive problems with primes
cites work: Sur l'hypothèse de Goldbach pour presque tous les nombres pairs
cites work: Q5769907
cites work: The binary Goldbach problem in short intervals. I: Explicit formula
cites work: The binary Goldbach problem in short intervals. II
cites work: Some remarks on Goldbach's problem
cites work: Some Results on Additive Number Theory
cites work: On Goldbach's Problem : Proof that Almost all Even Positive Integers are Sums of Two Primes
cites work: Q5815668
cites work: On a Problem of Additive Number Theory
cites work: The exceptional set of Goldbach's problem
cites work: Q4009400
cites work: On the Exceptional Set for Goldbach's Problem in Short Intervals
cites work: Über Zahlen, die sich als Summe einer Primzahl und einer 'kleinen' Potenz darstellen lassen. II
cites work: Über die Anwendung einer Methode von Linnik
cites work: Q4090431
cites work: Two remarks in prime number theory
cites work: Q5845915
cites work: Q5803461
cites work: Q5847301
cites work: Über das Primzahl-Zwillingsproblem. (On the twin-prime problem)
Recommended article: Q3703959
Recommended article: Goldbach numbers in sparse sequences
Recommended article: Q4384633
Recommended article: On the existence of a non-zero lower bound for the number of Goldbach partitions of an even integer
Recommended article: Refinements of Goldbach's conjecture, and the generalized Riemann hypothesis
Recommended article: New estimates for the exceptional set of the sum of two primes from an arithmetic progression
Recommended article: Q4252242
Recommended article: Q3803213
Recommended article: A remark on the Piatetski-Shapiro–Vinogradov theorem
Recommended article: Chen's theorem in totally real algebraic number fields
description: scientific article; zbMATH DE number 753671