Log-Like Functions and Uniform Distribution Modulo One

Rehberg, Martin

doi:10.2478/udt-2018-0013

.blurhash-client-img { display: none !important; }

Log-Like Functions and Uniform Distribution Modulo One

Uniform distribution theory

Volume 13 (2018): Issue 2 (December 2018)

By: Martin Rehberg

Open Access

|Jan 2019

Abstract

For a function f satisfying f (x) = o((log x) ^K), K > 0, and a sequence of numbers (q_n) _n, we prove by assuming several conditions on f that the sequence (αf (q_n)) _n≥n₀ is uniformly distributed modulo one for any nonzero real number α. This generalises some former results due to Too, Goto and Kano where instead of (q_n) _n the sequence of primes was considered.

References

[1] GOTO, K.—KANO, T.: Uniform Distribution of Some Special Sequences, Proc. Japan Acad. Ser. A 61 (1985), no. 3, 83–86.10.3792/pjaa.61.83
Search in Google Scholar Back to article
[2] _____Remarks to our Former Paper Uniform Distribution of Some Special Sequences, Proc. Japan Acad. Ser. A 68 (1992), no. 10, 348–350.10.3792/pjaa.68.348
Search in Google Scholar Back to article
[3] GROSSWALD, E.—SCHNITZER, F.J.: A Class of modified ζ and L-Functions, Pacific Journal of Mathematics 74 (1978), no. 2, 357–364.10.2140/pjm.1978.74.357
Search in Google Scholar Back to article
[4] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. Dover Publications, New York 2006.
Search in Google Scholar Back to article
[5] MÜLLER, H.: Über eine Klasse modifizierter ζ- und L-Funktionen, Arch. Math. 36 (1981), 157–161.10.1007/BF01223684
Search in Google Scholar Back to article
[6] STRAUCH, O.—PORUBSKÝ, Š.: Distribution of Sequences: A Sampler. eBook, 2016. https://math.boku.ac.at/udt/books/MYBASISNew.pdf
Search in Google Scholar Back to article
[7] TOO, Y.-H.: On the Uniform Distribution Modulo One of Some Log-like Sequences, Proc. Japan Acad. Ser. A 68 (1992), no. 9, 269–272.10.3792/pjaa.68.269
Search in Google Scholar Back to article
[8] PARENT, D. P.: Exercises in Number Theory. Springer, New York 1984.
Search in Google Scholar Back to article
[9] TITCHMARSH, E. C.: The Theory of the Riemann Zeta-Function. Oxford University Press, second edition, New York 1986.
Search in Google Scholar Back to article
[10] WINTNER, A.: On the cyclical distribution of the logarithms of the prime numbers, Quart. J. Math. 6 (1935), no. 1, 65–68.10.1093/qmath/os-6.1.65
Search in Google Scholar Back to article
[11] ZYGMUND, A.: Trigonometric Series. Cambridge University Press, third edition, London 2002.
Search in Google Scholar Back to article

DOI: https://doi.org/10.2478/udt-2018-0013 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

Journal RSS Feed

Language: English

Page range: 93 - 101

Submitted on: Oct 20, 2017

Accepted on: Feb 2, 2018

Published on: Jan 25, 2019

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

Uniform Distribution,

Discrepancy

Related subjects:

Mathematics,

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2019 Martin Rehberg, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Previous article Volume 13 (2018): Issue 2 (December 2018)Next article