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

Alessandro Languasco; Alberto Perelli

Annales de l'institut Fourier (1994)

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

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topLanguasco, 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 $\epsilon >0$, 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 $\epsilon >0$, 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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- [2] D.A. GOLDSTON, Linnik's theorem on Goldbach numbers in short intervals, Glasgow Math. J., 32 (1990), 285-297. Zbl0719.11065MR91i:11134
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- [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] 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] H. L. MONTGOMERY, R. C. VAUGHAN, The exceptional set in Goldbach's problem, Acta Arith., 27 (1975), 353-370. Zbl0301.10043MR51 #10263
- [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] 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] I.M. VINOGRADOV, Selected Works, Springer Verlag, 1985.

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