关于Ahmes级数的几个无理性问题

IF 0.6 3区 数学 Q3 MATHEMATICS
V. Kovač, T. Tao
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引用次数: 0

摘要

利用数学分析和初等概率论的基本工具,我们解决了一系列不同单位分数的无理性的几个问题,\(\sum_k 1/a_k\)。特别地,我们研究了Lambert级数\(\sum_k 1/(t^k-1)\)的子级数和PaulErdős和Ronald Graham引入的两类非理性序列\((a_k)\)。接下来,我们处理Erdős的问题,他问如果级数\(\sum_k 1/a_k\)和\(\sum_k 1/(a_k+1)\)都有有理数和,一个正整数序列\((a_k)\)能增长多快。我们构造的具有此性质的双指数增长序列\((a_k)\)推广到级数\(\sum_k 1/(a_k+j)\), \(j=0,1,2,\ldots,d-1\)的任意数\(d\),特别地,也给出了Erdős和Ernst Straus关于它们的和的\(d\) -元组集合的内部的一个正答案。最后,我们证明了一个数列\((a_k)\)的存在性,使得所有定义良好的和\(\sum_k 1/(a_k+t)\), \(t\in\mathbb{Z}\)都是有理数,从而否定了Kenneth Stolarsky的一个猜想。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On several irrationality problems for Ahmes series

Using basic tools of mathematical analysis and elementary probability theory we address several problems on the irrationality of series of distinct unit fractions,\(\sum_k 1/a_k\). In particular, we study subseries of the Lambert series \(\sum_k 1/(t^k-1)\) and two types of irrationality sequences \((a_k)\) introduced by Paul Erdős and Ronald Graham. Next, we address a question of Erdős, who asked how rapidly a sequence of positive integers \((a_k)\) can grow if both series \(\sum_k 1/a_k\) and \(\sum_k 1/(a_k+1)\)have rational sums. Our construction of double exponentially growing sequences \((a_k)\) with this property generalizes to any number \(d\) of series\(\sum_k 1/(a_k+j)\),\(j=0,1,2,\ldots,d-1\),and, in particular, also gives a positive answer to a question of Erdős and Ernst Straus on the interior of the set of \(d\)-tuples of their sums. Finally, we prove the existence of a sequence \((a_k)\) such that all well-defined sums \(\sum_k 1/(a_k+t)\),\(t\in\mathbb{Z}\), are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.

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来源期刊
CiteScore
1.50
自引率
11.10%
发文量
77
审稿时长
4-8 weeks
期刊介绍: Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.
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