zbMATH — the first resource for mathematics

On Linnik’s theorem on Goldbach numbers in short intervals and related problems. (English) Zbl 0799.11040
Linnik proved, assuming the Riemann hypothesis, that for any \(\varepsilon>0\), the interval \([N, N+\log^{3+\varepsilon} N]\) 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 forms of Parseval’s identity for exponential sums over primes.
Reviewer: A.Perelli (Genova)

11P32 Goldbach-type theorems; other additive questions involving primes
11L20 Sums over primes
11L07 Estimates on exponential sums
Full Text: DOI Numdam EuDML
[1] P.X. GALLAGHER, Some consequences of the Riemann hypothesis, Acta Arith., 37 (1980), 339-343. · Zbl 0444.10034
[2] D.A. GOLDSTON, Linnik’s theorem on Goldbach numbers in short intervals, Glasgow Math. J., 32 (1990), 285-297. · Zbl 0719.11065
[3] H. HALBERSTAM, H.-E. RICHERT, Sieve methods, Academic Press, 1974. · Zbl 0298.10026
[4] I. KÁTAI, A remark on a paper of ju. V. linnik (Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., 17 (1967), 99-100. · Zbl 0145.04905
[5] A.F. LAVRIK, Estimation of certain integrals connected with the additive problems (Russian), Vestnik Leningrad Univ., 19 (1959), 5-12. · Zbl 0092.04401
[6] Yu. V. LINNIK, Some conditional theorems concerning the binary Goldbach problem (Russian), Izv. Akad. Nauk SSSR, Ser. Mat., 16 (1952), 503-520. · Zbl 0049.03104
[7] H. L. MONTGOMERY, R. C. VAUGHAN, The exceptional set in Goldbach’s problem, Acta Arith., 27 (1975), 353-370. · Zbl 0301.10043
[8] B. SAFFARI, R. C. VAUGHAN, On the fractional parts of x/n and related sequences II, Ann. Inst. Fourier, 27-2 (1977), 1-30. · Zbl 0379.10023
[9] A. SELBERG, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid., 47 (1943), 87-105. · Zbl 0063.06869
[10] I.M. VINOGRADOV, Selected works, Springer Verlag, 1985.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.