[La conjecture de Goldbach et le plus petit nombre premier dans une progression arithmétique]
Dans ce document, nous essayons d'étudier les relations entre la conjecture de Goldbach et le plus petit nombre premier dans une progression arithmétique. Nous donnons une nouvelle forme faible de la conjecture de Goldbach. Nous prouvons que cette forme affaiblie et une forme affaiblie de l'hypothèse de Chowla impliquent que tout entier pair suffisamment grand peut être écrit comme une somme de deux nombres premiers distincts.
In this Note, we try to study the relations between the Goldbach Conjecture and the least prime number in an arithmetic progression. We give a new weakened form of the Goldbach Conjecture. We prove that this weakened form and a weakened form of the Chowla Hypothesis imply that every sufficiently large even integer may be written as the sum of two distinct primes.
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@article{CRMATH_2010__348_5-6_241_0,
author = {Zhang, Shaohua},
title = {Goldbach {Conjecture} and the least prime number in an arithmetic progression},
journal = {Comptes Rendus. Math\'ematique},
pages = {241--242},
publisher = {Elsevier},
volume = {348},
number = {5-6},
year = {2010},
doi = {10.1016/j.crma.2010.02.011},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.02.011/}
}
TY - JOUR AU - Zhang, Shaohua TI - Goldbach Conjecture and the least prime number in an arithmetic progression JO - Comptes Rendus. Mathématique PY - 2010 SP - 241 EP - 242 VL - 348 IS - 5-6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.02.011/ DO - 10.1016/j.crma.2010.02.011 LA - en ID - CRMATH_2010__348_5-6_241_0 ER -
%0 Journal Article %A Zhang, Shaohua %T Goldbach Conjecture and the least prime number in an arithmetic progression %J Comptes Rendus. Mathématique %D 2010 %P 241-242 %V 348 %N 5-6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.02.011/ %R 10.1016/j.crma.2010.02.011 %G en %F CRMATH_2010__348_5-6_241_0
Zhang, Shaohua. Goldbach Conjecture and the least prime number in an arithmetic progression. Comptes Rendus. Mathématique, Tome 348 (2010) no. 5-6, pp. 241-242. doi : 10.1016/j.crma.2010.02.011. http://www.numdam.org/articles/10.1016/j.crma.2010.02.011/
[1] On the least prime in an arithmetic progression, J. Indian Math. Soc., Volume 1 (1934) no. 2, pp. 1-3
[2] Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3), Volume 64 (1992) no. 2, pp. 265-338
Cité par Sources :
☆ This work was partially supported by the National Basic Research Program (973) of China (No. 2007CB807902) and the Natural Science Foundation of Shandong Province (No. Y2008G23).
