Linnik proved, assuming the Riemann Hypothesis, that for any , the interval contains a number which is the sum of two primes, provided that is sufficiently large. This has subsequently been improved to the same assertion being valid for the smaller gap , the added new ingredient being Selberg’s estimate for the mean-square of primes in short intervals. Here we give another proof of this sharper result which avoids the use of Selberg’s estimate and is therefore more in the spirit of Linnik’s original approach. We also improve an unconditional result of Lavrik’s on truncated froms of Parseval’s identity for exponential sums over primes.
En admettant l’hypothèse de Riemann, Linnik a prouvé que, pour tout et pour assez grand, l’intervalle contient un entier qui est somme de deux nombres premiers. Ce résultat a été amélioré ensuite en prouvant que la propriété reste vraie pour l’écart , en utilisant l’estimation de Selberg pour la moyenne quadratique des nombres premiers dans les petits intervalles. On donne ici une nouvelle démonstration du deuxième résultat qui, n’utilisant pas l’estimation de Selberg, suit davantage l’esprit de l’approche originale de Linnik. On améliore aussi un résultat de Lavrik concernant des formes tronquées de l’identité de Parseval pour des sommes d’exponentielles sur les nombres premiers.
@article{AIF_1994__44_2_307_0, author = {Languasco, Alessandro and Perelli, Alberto}, title = {On {Linnik's} theorem on {Goldbach} numbers in short intervals and related problems}, journal = {Annales de l'Institut Fourier}, pages = {307--322}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {44}, number = {2}, year = {1994}, doi = {10.5802/aif.1399}, mrnumber = {95g:11097}, zbl = {0799.11040}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1399/} }
TY - JOUR AU - Languasco, Alessandro AU - Perelli, Alberto TI - On Linnik's theorem on Goldbach numbers in short intervals and related problems JO - Annales de l'Institut Fourier PY - 1994 SP - 307 EP - 322 VL - 44 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1399/ DO - 10.5802/aif.1399 LA - en ID - AIF_1994__44_2_307_0 ER -
%0 Journal Article %A Languasco, Alessandro %A Perelli, Alberto %T On Linnik's theorem on Goldbach numbers in short intervals and related problems %J Annales de l'Institut Fourier %D 1994 %P 307-322 %V 44 %N 2 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1399/ %R 10.5802/aif.1399 %G en %F AIF_1994__44_2_307_0
Languasco, Alessandro; Perelli, Alberto. On Linnik's theorem on Goldbach numbers in short intervals and related problems. Annales de l'Institut Fourier, Volume 44 (1994) no. 2, pp. 307-322. doi : 10.5802/aif.1399. http://archive.numdam.org/articles/10.5802/aif.1399/
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