Short intervals almost all containing primes

N. Watt

Acta Arithmetica (1995)

  • Volume: 72, Issue: 2, page 131-167
  • ISSN: 0065-1036

How to cite

top

N. Watt. "Short intervals almost all containing primes." Acta Arithmetica 72.2 (1995): 131-167. <http://eudml.org/doc/206789>.

@article{N1995,
author = {N. Watt},
journal = {Acta Arithmetica},
keywords = {primes; short interval; sieve method; mean value},
language = {eng},
number = {2},
pages = {131-167},
title = {Short intervals almost all containing primes},
url = {http://eudml.org/doc/206789},
volume = {72},
year = {1995},
}

TY - JOUR
AU - N. Watt
TI - Short intervals almost all containing primes
JO - Acta Arithmetica
PY - 1995
VL - 72
IS - 2
SP - 131
EP - 167
LA - eng
KW - primes; short interval; sieve method; mean value
UR - http://eudml.org/doc/206789
ER -

References

top
  1. [1] R. C. Baker and G. Harman, The difference between consecutive primes, preprint. Zbl0853.11076
  2. [2] A. Buchstab, Asymptotic estimates of a general number-theoretic function, Mat. Sb. (N.S.) (2) 44 (1937), 1239-1246 (in Russian with a German summary). 
  3. [3] H. Davenport, Multiplicative Number Theory, Springer, 1980. 
  4. [4] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982), 219-288. 
  5. [5] J.-M. Deshouillers and H. Iwaniec, Power mean values of the Riemann zeta-function, Mathematika 29 (1982), 202-212. 
  6. [6] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1954. Zbl0058.03301
  7. [7] G. Harman, Almost-primes in short intervals, Math. Ann. 258 (1981), 107-112. Zbl0474.10034
  8. [8] G. Harman, Primes in short intervals, Math. Z. 180 (1982), 335-348. Zbl0482.10040
  9. [9] G. Harman, On the distribution of αp modulo one, J. London Math. Soc. (2) 27 (1983), 9-18. Zbl0504.10018
  10. [10] G. Harman, On the distribution of αp modulo one II, preprint. 
  11. [11] D. R. Heath-Brown, Gaps between primes and the pair correlation of zeros of the zeta-function, Acta Arith. 41 (1982), 85-99. Zbl0414.10044
  12. [12] D. R. Heath-Brown, Finding primes by sieve methods, Proc. 1982 ICM, Warsaw, 1983, PWN, Vol. 1, Warszawa, 1984, 487-492. Zbl0565.10039
  13. [13] D. R. Heath-Brown and H. Iwaniec, On the difference between consecutive primes, Invent. Math. 55 (1979), 49-69. Zbl0424.10028
  14. [14] M. N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164-170. Zbl0241.10026
  15. [15] H. Iwaniec, Rosser's sieve, Acta Arith. 36 (1980), 171-202. 
  16. [16] H. Iwaniec, A new form of the error term in the linear sieve, Acta Arith. 37 (1980), 307-320. Zbl0444.10038
  17. [17] H. Iwaniec and M. Jutila, Primes in short intervals, Ark. Mat. 17 (1979), 167-176. Zbl0408.10029
  18. [18] H. Iwaniec and J. Pintz, Primes in short intervals, Monatsh. Math. 98 (1984), 115-143. Zbl0544.10040
  19. [19] C. Jia, On the Goldbach numbers in the short interval, Science in China, to appear. 
  20. [20] C. Jia, On the difference between consecutive primes, Science in China,, to appear. 
  21. [21] H. Li, Primes in short intervals, unpublished manuscript. Zbl0875.11016
  22. [22] H. Li, Primes in short intervals, preprint. Zbl0875.11016
  23. [23] S. Lou and Q. Yao, A Chebychev's type of prime number theorem in a short interval - II, Hardy-Ramanujan J. 15 (1992), 1-33. Zbl0780.11039
  24. [24] H. Mikawa, Almost-primes in arithmetic progressions and short intervals, Tsukuba J. Math. (2) 13 (1989), 387-401. Zbl0689.10052
  25. [25] H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer, 1971. Zbl0216.03501
  26. [26] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc. (2) 8 (1974), 73-82. Zbl0281.10021
  27. [27] H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353-370. Zbl0301.10043
  28. [28] Y. Motohashi, A note on almost-primes in short intervals, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), 225-226. Zbl0445.10032
  29. [29] C. J. Mozzochi, On the difference between consecutive primes, J. Number Theory 24 (1986), 181-187. 
  30. [30] A. Perelli and J. Pintz, On the exceptional set for Goldbach's Problem in short intervals, J. London Math. Soc. (2) 47 (1993), 41-49. Zbl0806.11042
  31. [31] A. Selberg, On the normal density of primes in short intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), 87-105. Zbl0063.06869
  32. [32] P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math. 313 (1980), 161-170. Zbl0412.10030
  33. [33] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford, 1986. Zbl0601.10026
  34. [34] N. Watt, Kloosterman sums and a mean value for Dirichlet polynomials, J. Number Theory, to appear. Zbl0837.11050
  35. [35] D. Wolke, Fast-Primzahlen in kurzen Intervallen, Math. Ann. 224 (1979), 233-242 Zbl0402.10042

NotesEmbed ?

top

You must be logged in to post comments.