<abstract xmlns="http://www.w3.org/1999/xhtml">It is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (<italic>e</italic>2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function <italic>g</italic> with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number <italic>α</italic> and almost all real numbers <italic>β></italic> 1 (alternatively, for a fixed real number <italic>β></italic> 1 and almost all real numbers <italic>α</italic>) and for all real polynomials <italic>Q</italic>(<italic>x</italic>), sequences (<italic>αβ</italic><italic>n</italic><italic>g</italic>(<italic>β</italic>)+ <italic>Q</italic>(<italic>n</italic>)) n∈𝕅 are uniformly distributed modulo 1.</abstract>

It is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

Higher Order Oscillation and Uniform Distribution

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Higher Order Oscillation and Uniform Distribution"}

It is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating...

Higher Order Oscillation and Uniform Distribution

Publié en ligne: 27 mars 2020

Pages: 1 - 10

Reçu: 25 avr. 2017

Accepté: 09 avr. 2018

DOI: https://doi.org/10.2478/udt-2019-0001

Mots clés
higher order oscillating sequence, uniformly distributed modulo 1 (u. d. mod 1)

© 2019 Shigeki Akiyama et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Higher Order Oscillation and Uniform Distribution

Publié en ligne: 27 mars 2020

Pages: 1 - 10

Reçu: 25 avr. 2017

Accepté: 09 avr. 2018

DOI: https://doi.org/10.2478/udt-2019-0001

Mots cléshigher order oscillating sequence, uniformly distributed modulo 1 (u. d. mod 1)

© 2019 Shigeki Akiyama et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mots clés
higher order oscillating sequence, uniformly distributed modulo 1 (u. d. mod 1)