arXiv subjects

Louna Seppälä

Publications and source records attributed to Louna Seppälä.

6 recordsLinked to original sources

arXiv2022

Euler's factorial series, Hardy integral, and continued fractions

We study $p$-adic Euler's series $E_p(t) = \sum_{k=0}^{\infty}k!t^k$ at a point $p^a$, $a \in \mathbb{Z}_{\ge 1}$, and use Padé approximations to prove a lower bound for the $p$-adic absolute value of the expression $cE_p\left(\pm p^a\right)-d$, where $c, d \in \mathbb{Z}$. It is interesting that the same Padé polynomials which $p$-adically converge to $E_p(t)$, approach the Hardy integral $\mathcal{H}(t) = \int_{0}^{\infty} \frac{e^{-s}}{1-ts}ds$ on the Archimedean side. This connection is used with a trick of analytic continuation when deducing an Archimedean bound for the numerator Padé polynomial needed in the derivation of the lower bound for $|cE_p\left(\pm p^a\right)-d|_p$. Furthermore, we present an interconnection between $E(t)$ and $\mathcal{H}(t)$ via continued fractions.

math.NT↗

arXiv2022

A $p$-adic lower bound for a linear form in logarithms

Linear forms in logarithms have an important role in the theory of Diophantine equations. In this article, we prove explicit $p$-adic lower bounds for linear forms in $p$-adic logarithms of rational numbers using Padé approximations of the second kind.

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arXiv2018

Euler's factorial series at algebraic integer points

We study a linear form in the values of Euler's series $F(t)=\sum_{n=0}^\infty n!t^n$ at algebraic integer points $α_1, \ldots, α_m \in \mathbb{Z}_{\mathbb{K}}$ belonging to a number field $\mathbb{K}$. Let $v|p$ be a non-Archimedean valuation of $\mathbb{K}$. Two types of non-vanishing results for the linear form $Λ_v = λ_0 + λ_1 F_v(α_1) + \ldots + λ_m F_v(α_m)$, $λ_i \in \mathbb{Z}_{\mathbb{K}}$, are derived, the second of them containing a lower bound for the $v$-adic absolute value of $Λ_v$. The first non-vanishing result is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Padé approximations to the generalised factorial series $\sum_{n=0}^\infty \left( \prod_{k=0}^{n-1} P(k) \right) t^n$, where $P(x)$ is a polynomial of degree one.

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arXiv2018

Euler's divergent series in arithmetic progressions

Let $ξ$ and $m$ be integers satisfying $ξ\ne 0$ and $m\ge 3$. We show that for any given integers $a$ and $b$, $b \neq 0$, there are $\frac{φ(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$ such that $a-bF_p(ξ) \ne 0$, where $F_p(ξ)=\sum_{n=0}^\infty n!ξ^n$ is the $p$-adic evaluation of Euler's factorial series at the point $ξ$.

math.NT↗

arXiv2018

On Mahler's transcendence measure for $e$

We present a completely explicit transcendence measure for $e$. This is a continuation and an improvement to the works of Borel, Mahler and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of $e$. The results are based on Hermite-Padé approximations and on careful analysis of common factors in the footsteps of Hata.

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arXiv2018

Hermite-Thue equation: Padé approximations and Siegel's lemma

Padé approximations and Siegel's lemma are widely used tools in Diophantine approximation theory. This work has evolved from the attempts to improve Baker-type linear independence measures, either by using the Bombieri-Vaaler version of Siegel's lemma to sharpen the estimates of Padé-type approximations, or by finding completely explicit expressions for the yet unknown 'twin type' Hermite-Padé approximations. The appropriate homogeneous matrix equation representing both methods has an $M \times (L+1)$ coefficient matrix, where $M \le L$. The homogeneous solution vectors of this matrix equation give candidates for the Padé polynomials. Due to the Bombieri-Vaaler version of Siegel's lemma, the upper bound of the minimal non-zero solution of the matrix equation can be improved by finding the gcd of all the $M \times M$ minors of the coefficient matrix. In this paper we consider the exponential function and prove that there indeed exists a big common factor of the $M \times M$ minors, giving a possibility to apply the Bombieri-Vaaler version of Siegel's lemma. Further, in the case $M=L$, the existence of this common factor is a step towards understanding the nature of the 'twin type' Hermite-Padé approximations to the exponential function.

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