On Linnik's theorem on Goldbach numbers in short intervals and related problems

Languasco, Alessandro; Perelli, Alberto

Annales de l'Institut Fourier, Tome 44 (1994) no. 2, pp. 307-322.

En admettant l’hypothèse de Riemann, Linnik a prouvé que, pour tout $ϵ>0$ et pour $N$ assez grand, l’intervalle $[N,N+{log}^{3+ϵ}N]$ 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 $C{log}^{2}N$, 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.

Linnik proved, assuming the Riemann Hypothesis, that for any $ϵ>0$, the interval $[N,N+{log}^{3+ϵ}N]$ contains a number which is the sum of two primes, provided that $N$ is sufficiently large. This has subsequently been improved to the same assertion being valid for the smaller gap $C{log}^{2}N$, 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.

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