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

• Volume: 44, Issue: 2, page 307-322
• ISSN: 0373-0956

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## Abstract

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Linnik proved, assuming the Riemann Hypothesis, that for any $ϵ>0$, the interval $\left[N,N+{log}^{3+ϵ}N\right]$ 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 $C\phantom{\rule{0.166667em}{0ex}}{log}^{2}\phantom{\rule{0.166667em}{0ex}}N$, 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.

## How to cite

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Languasco, Alessandro, and Perelli, Alberto. "On Linnik's theorem on Goldbach numbers in short intervals and related problems." Annales de l'institut Fourier 44.2 (1994): 307-322. <http://eudml.org/doc/75063>.

@article{Languasco1994,
abstract = {Linnik proved, assuming the Riemann Hypothesis, that for any $\epsilon &gt;0$, the interval $[N,N+\log ^\{3+\epsilon \}N]$ 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.},
author = {Languasco, Alessandro, Perelli, Alberto},
journal = {Annales de l'institut Fourier},
keywords = {Linnik's theorem; Goldbach numbers in short intervals; Goldbach problem; Lavrik's result; Parseval's identity; exponential sums over primes},
language = {eng},
number = {2},
pages = {307-322},
publisher = {Association des Annales de l'Institut Fourier},
title = {On Linnik's theorem on Goldbach numbers in short intervals and related problems},
url = {http://eudml.org/doc/75063},
volume = {44},
year = {1994},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 2
SP - 307
EP - 322
AB - Linnik proved, assuming the Riemann Hypothesis, that for any $\epsilon &gt;0$, the interval $[N,N+\log ^{3+\epsilon }N]$ 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.
LA - eng
KW - Linnik's theorem; Goldbach numbers in short intervals; Goldbach problem; Lavrik's result; Parseval's identity; exponential sums over primes
UR - http://eudml.org/doc/75063
ER -

## References

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1. [1] P.X. GALLAGHER, Some consequences of the Riemann hypothesis, Acta Arith., 37 (1980), 339-343. Zbl0444.10034MR82j:10071
2. [2] D.A. GOLDSTON, Linnik's theorem on Goldbach numbers in short intervals, Glasgow Math. J., 32 (1990), 285-297. Zbl0719.11065MR91i:11134
3. [3] H. HALBERSTAM, H.-E. RICHERT, Sieve Methods, Academic Press, 1974. Zbl0298.10026MR54 #12689
4. [4] I. KÁTAI, A remark on a paper of Ju. V. Linnik (Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., 17 (1967), 99-100. Zbl0145.04905MR35 #5407
5. [5] A.F. LAVRIK, Estimation of certain integrals connected with the additive problems (Russian), Vestnik Leningrad Univ., 19 (1959), 5-12. Zbl0092.04401MR22 #7986
6. [6] Yu. V. LINNIK, Some conditional theorems concerning the binary Goldbach problem (Russian), Izv. Akad. Nauk SSSR, Ser. Mat., 16 (1952), 503-520. Zbl0049.03104
7. [7] H. L. MONTGOMERY, R. C. VAUGHAN, The exceptional set in Goldbach's problem, Acta Arith., 27 (1975), 353-370. Zbl0301.10043MR51 #10263
8. [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. Zbl0379.10023MR58 #554a
9. [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. Zbl0063.06869MR7,48e
10. [10] I.M. VINOGRADOV, Selected Works, Springer Verlag, 1985.

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