1. bookVolume 16 (2021): Issue 2 (December 2021)
Journal Details
License
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English
access type Open Access

Mahler’s Conjecture on ξ(3/2)nmod 1

Published Online: 02 Feb 2022
Volume & Issue: Volume 16 (2021) - Issue 2 (December 2021)
Page range: 49 - 70
Received: 27 Feb 2019
Accepted: 29 Aug 2021
Journal Details
License
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English
Abstract

K. Mahler’s conjecture: There exists no ξ ∈ ℝ+ such that the fractional parts {ξ(3/2)n} satisfy 0 {ξ(3/2)n} < 1/2 for all n = 0, 1, 2,... Such a ξ, if exists, is called a Mahler’s Z-number. In this paper we prove that if ξ is a Z-number, then the sequence xn = {ξ(3/2)n}, n =1, 2,... has asymptotic distribution function c0(x), where c0(x)=1 for x ∈ (0, 1].

Keywords

MSC 2010

MAHLER, K.: An unsolved problem on the powers of 3/2, J. Austral. Math. Soc. 8 (1968), 313–321.10.1017/S1446788700005371Search in Google Scholar

STRAUCH, O.: On distribution functions of ξ(3/2)n mod 1, Acta Arith. 81 (1997), no. 1, 25–35.Search in Google Scholar

YOUNG, L. C.: General inequalities of Stieltjes integrals and the convergence of Fourier series, Math. Ann. 115 (1938), no. 1, 581–612.Search in Google Scholar

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