Irregularities in the Distribution of Prime Numbers in a Beatty Sequence
Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 738-743

Voir la notice de l'article provenant de la source Cambridge University Press

We prove irregularities in the distribution of prime numbers in any Beatty sequence ${\mathcal{B}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ is a positive real irrational number of finite type.
DOI : 10.4153/S0008439519000778
Mots-clés : prime numbers, irregularities, Maier’s matrix method, Beatty sequence
Tongsomporn, Janyarak; Steuding, Jörn. Irregularities in the Distribution of Prime Numbers in a Beatty Sequence. Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 738-743. doi: 10.4153/S0008439519000778
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[1] Banks, W. D. and Shparlinski, I. E., Prime numbers with Beatty sequences. Colloq. Math. 115(2009), 147–157. https://doi.org/10.4064/cm115-2-1 Google Scholar | DOI

[2] Buchstab, A. A., Asymptotic estimates of a general number-theoretic function. Mat.-Sb. (N.S.) 2(1937), 1239–1246. Google Scholar

[3] Cramér, H., On the order of magnitude of the difference between consecutive prime numbers. Acta Arith. 2(1936), 23–46. Google Scholar | DOI

[4] Davenport, H., Multiplicative number theory, Third ed, Graduate Texts in Mathematics, 74, Springer-Verlag, New York, 2000. Google Scholar

[5] Fraenkel, A. S., The bracket function and complementary sets of integers. Canad. J. Math. 21(1969), 6–27. https://doi.org/10.4153/CJM-1969-002-7 Google Scholar | DOI

[6] Gallagher, P. X., A large sieve density estimate near 𝜎 = 1. Invent. Math. 11(1970), 329–339. https://doi.org/10.1007/BF01403187 Google Scholar | DOI

[7] Granville, A., Unexpected irregularities in the distribution of prime numbers. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994). Birkhäuser, Basel, 1995, pp. 388–399. Google Scholar | DOI

[8] Hildebrand, A. and Maier, H., Irregularities in the distribution of primes in short intervals. J. Reine Angew. Math. 397(1989), 162–193. Google Scholar

[9] Iwaniec, H., The sieve of Eratosthenes-Legendre. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(1977), 257–268. Google Scholar

[10] Maier, H., Chains of large gaps between consecutive primes. Adv. in Math. 39(1981), 257–269. https://doi.org/10.1016/0001-8708(81)90003-7 Google Scholar | DOI

[11] Maier], H., Primes in short intervals. Michigan Math. J. 32(1985), 221–225. https://doi.org/10.1307/mmj/1029003189 Google Scholar

[12] Selberg, A., On the normal density of primes in small intervalls and the difference between conscutive primes. Arch. Math. Naturvid. 47(1943), 87–105. Google Scholar

[13] Tadee, S. and Laohakosol, V., Complementary sets and Beatty functions. Thai J. Math. 3(2005), 27–33. https://doi.org/10.1155/2005/165785 Google Scholar

[14] Thorne, F., Maier matrices beyond Z.. In: Combinatorial number theory. Walter de Gruyter, Berlin, 2009, pp. 185–192. Google Scholar

[15] Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers. Dover Publications, Mineola, NY, 2004. Google Scholar

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