# Goldbach’s problem with primes in arithmetic progressions and in short intervals

• [1] Universität Münster Mathematisches Institut Einsteinstr. 62 D-48149 Münster, Germany
• Volume: 25, Issue: 2, page 331-351
• ISSN: 1246-7405

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## Abstract

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Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd $n¬\equiv 1\phantom{\rule{0.222222em}{0ex}}\left(6\right)$, Goldbach’s ternary problem $n={p}_{1}+{p}_{2}+{p}_{3}$ is solvable with primes ${p}_{1},{p}_{2}$ in short intervals ${p}_{i}\in \left[{X}_{i},{X}_{i}+Y\right]$ with ${X}_{i}^{{\theta }_{i}}=Y$, $i=1,2$, and ${\theta }_{1},{\theta }_{2}\ge 0.933$ such that $\left({p}_{1}+2\right)\left({p}_{2}+2\right)$ has at most $9$ prime factors.

## How to cite

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Halupczok, Karin. "Goldbach’s problem with primes in arithmetic progressions and in short intervals." Journal de Théorie des Nombres de Bordeaux 25.2 (2013): 331-351. <http://eudml.org/doc/275735>.

@article{Halupczok2013,
abstract = {Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd $n\lnot \equiv 1\:(6)$, Goldbach’s ternary problem $n=p_1+p_2+p_3$ is solvable with primes $p_\{1\},p_\{2\}$ in short intervals $p_i \in [X_i,X_i+Y]$ with $X_\{i\}^\{\theta _\{i\}\}=Y$, $i=1,2$, and $\theta _\{1\},\theta _\{2\}\ge 0.933$ such that $(p_\{1\}+2)(p_\{2\}+2)$ has at most $9$ prime factors.},
affiliation = {Universität Münster Mathematisches Institut Einsteinstr. 62 D-48149 Münster, Germany},
author = {Halupczok, Karin},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {additive problems; circle method; sieve methods; short intervals; short interval},
language = {eng},
month = {9},
number = {2},
pages = {331-351},
publisher = {Société Arithmétique de Bordeaux},
title = {Goldbach’s problem with primes in arithmetic progressions and in short intervals},
url = {http://eudml.org/doc/275735},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Halupczok, Karin
TI - Goldbach’s problem with primes in arithmetic progressions and in short intervals
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/9//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 2
SP - 331
EP - 351
AB - Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd $n\lnot \equiv 1\:(6)$, Goldbach’s ternary problem $n=p_1+p_2+p_3$ is solvable with primes $p_{1},p_{2}$ in short intervals $p_i \in [X_i,X_i+Y]$ with $X_{i}^{\theta _{i}}=Y$, $i=1,2$, and $\theta _{1},\theta _{2}\ge 0.933$ such that $(p_{1}+2)(p_{2}+2)$ has at most $9$ prime factors.
LA - eng
KW - additive problems; circle method; sieve methods; short intervals; short interval
UR - http://eudml.org/doc/275735
ER -

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