arXiv subjects

Yohei Tachiya

Publications and source records attributed to Yohei Tachiya.

8 recordsLinked to original sources

arXiv2026

Refinements of Erd\H{o}s's irrationality criterion for certain sparse infinite series

In this paper, we establish new irrationality criteria for certain sparse power series. As applications of these criteria, we generalize a result of Erd\H{o}s and obtain several irrationality results for various infinite series involving the classical arithmetic functions. For example, we prove that for any integers $t\ge2$ and $k\geq0$, the numbers \[ \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{\sigma(n)}} \quad\text{and}\quad \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{\phi(n)}} \] are both irrational, where $d(n)$, $\sigma(n)$, and $\phi(n)$ denote the number of divisors, the sum of divisors, and Euler's totient functions, respectively.

math.NT↗

arXiv2025

On the number of $k$-full integers between three successive $k$-th powers

Let $k\geq2$ be an integer. The aim of this paper is to investigate the distribution of $k$-full integers between three successive $k$-th powers. More precisely, for any integers $\ell,m\ge0$, we establish the explicit asymptotic density for the set of integers $n$ such that the intervals $(n^k, (n+1)^k)$ and $((n+1)^k, (n+2)^k)$ contain exactly $\ell$ and $m$ $k$-full integers, respectively. As an application, we prove that there are infinitely many triples of successive $k$-th powers in the sequence of $k$-full integers, thereby providing a more general answer to Shiu's question.

math.NT↗

arXiv2023

Linear independence of series related to the Thue--Morse sequence along powers

The Thue--Morse sequence $\{t(n)\}_{n\geqslant 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$, where $t(n)=1$ (resp. $=0$) if the binary expansion of $n$ has an odd (resp. even) number of ones. In this paper, we generalize a recent result of E.~Miyanohara by showing that, for a fixed Pisot or Salem number $β>\sqrtφ=1.272019649\ldots$, the set of the numbers $$ 1,\quad \sum_{n\geqslant 1}\frac{t(n)}{β^{n}},\quad \sum_{n\geqslant 1}\frac{t(n^2)}{β^{n}},\quad \dots, \quad \sum_{n\geqslant 1}\frac{t(n^k)}{β^{n}},\quad \dots $$ is linearly independent over the field $\mathbb{Q}(β)$, where $φ:=(1+\sqrt{5})/2$ is the golden ratio. Our result implies that for any $k\geqslant 1$ and for any $a_1,a_2,\ldots,a_k\in\mathbb{Q}(β)$, not all zero, the sequence \{$a_1t(n)+a_2t(n^2)+\cdots+a_kt(n^k)\}_{n\geqslant 1}$ cannot be eventually periodic.

math.NT↗

arXiv2022

Linear independence of certain numbers in the base-$b$ number system

Let $(i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}$ and $S_{i,j}$ be an infinite subset of positive integers including all prime numbers in some arithmetic progression. In this paper, we prove the linear independence over $\mathbb{Q}$ of the numbers \[ 1, \quad \sum_{n\in S_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}},\quad (i,j)\in \mathbb{N}\times \mathbb{N}_{\geq2}, \] where $b\geq2$ is an integer and $a_{i,j}(n)$ are bounded nonzero integer-valued functions on $S_{i,j}$. Moreover, we also establish a necessary and sufficient condition on the subset $\mathcal{A}$ of $\mathbb{N}\times \mathbb{N}_{\geq2}$ for the numbers \[ 1, \quad \sum_{n\in T_{i,j}}^{}\frac{a_{i,j}(n)}{b^{in^j}},\quad (i,j)\in \mathcal{A} \] to be linearly independent over $\mathbb{Q}$ for any given infinite subsets $T_{i,j}$ of positive integers. Our theorems generalize a result of V. Kumar.

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arXiv2020

Algebraic independence of certain infinite products involving the Fibonacci numbers

Let $\{F_{n}\}_{n\geq0}$ be the sequence of the Fibonacci numbers. The aim of this paper is to give explicit formulae for the infinite products \[ \prod_{n=1}^{\infty}\left( 1+\frac{1}{F_{n}}\right) ,\qquad\prod_{n=3}^{\infty}\left( 1-\frac{1}{F_{n}}\right) \] in terms of the values of the Jacobi theta functions. From this we deduce the algebraic independence over $\mathbb{Q}$ of the above numbers by applying Bertrand's theorem on the algebraic independence of the values of the Jacobi theta functions.

math.NT↗

arXiv2019

Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers

The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[ 1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}} \] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and $\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.

math.NT↗

arXiv2016

Algebraic independence results for values of Jacobi theta-constants

Let $θ_3(τ)=1+2\sum_{ν=1}^{\infty} q^{ν^2}$ with $q=e^{iπτ}$ and $\Im (τ)>0$ denote the Thetanullwert of the Jacobi theta function \[θ(z|τ) \,=\,\sum_{ν=-\infty}^{\infty} e^{πiν^2τ+ 2πiνz} \,.\] Moreover, let $θ_2(τ)=2\sum_{ν=0}^{\infty} q^{{(ν+1/2)}^2}$ and $θ_4(τ)=1+2\sum_{ν=1}^{\infty} {(-1)}^νq^{ν^2}$. For every even integer $n\geq 6$, which is not a power of two, we prove constructively the existence of a nontrivial integer polynomial $Q_n(X,Y)$ such that \[Q_n\Big( \,\frac{θ_3^4(nτ)}{θ_3^4(τ)},\frac{θ_2^4(τ)}{θ_3^4(τ)}\, \Big) \,=\, 0 \] holds for all complex numbers $τ$ from the upper half plane of $\mathbb{C}$. These polynomials are used to prove the algebraic independence of $θ_3(nτ)$ and $θ_3(τ)$ for all algebraic numbers $q=e^{iπτ}$ with $0<|q|<1$. Combining this with former results of the authors, it is shown that for such algebraic $q$ the numbers $θ_3(nτ)$ and $θ_3(τ)$ are algebraically independent over $\mathbb{Q}$ for every integer $n\geq 2$. A result on the algebraic dependence over $\mathbb{Q}$ of the three numbers $θ_3(\ellτ)$, $θ_3(mτ)$, and $θ_3(nτ)$ for integers $\ell,m,n\geq 1$ is also presented.

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arXiv2016

Algebraic independence results for values of Theta-constants, II

Let $θ_3(τ)=1+2\sum_{ν=1}^{\infty} q^{ν^2}$ with $q=e^{iπτ}$ denote the Thetanullwert of the Jacobi theta function \[θ(z|τ) \,=\,\sum_{ν=-\infty}^{\infty} e^{πiν^2τ+ 2πiνz} \,.\] Moreover, let $θ_2(τ)=2\sum_{ν=0}^{\infty} q^{{(ν+1/2)}^2}$ and $θ_4(τ)=1+2\sum_{ν=1}^{\infty} {(-1)}^νq^{ν^2}$. For algebraic numbers $q$ with $0<|q|<1$ and for any $j\in \{ 2,3,4\}$ we prove the algebraic independence over $\mathbb{Q}$ of the numbers $θ_j(nτ)$ and $θ_j(τ)$ for all odd integers $n\geq 3$. Assuming the same conditions on $q$ and $τ$ as above, we obtain sufficient conditions by use of a criterion involving resultants in order to decide on the algebraic independence over $\mathbb{Q}$ of $θ_j(2mτ)$ and $θ_j(τ)$ $(j=2,3,4)$ and of $θ_3(4mτ)$ and $θ_3(τ)$ with odd positive integers $m$. In particular, we prove the algebraic independence of $θ_3(nτ)$ and $θ_3(τ)$ for even integers $n$ with $2\leq n\leq 22$. The paper continues the work of the first-mentioned author, who already proved the algebraic independence of $θ_3(2^mτ)$ and $θ_3(τ)$ for $m=1,2,\dots$.

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