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Anne-Maria Ernvall-Hytönen

Publications and source records attributed to Anne-Maria Ernvall-Hytönen.

At least 19 recordsLinked to original sources
arXiv2025

Explicit estimates for the Goldbach summatory function

In order to study the analytic properties of the Goldbach generating function we consider a smooth version, similar to the Chebyshev function for the Prime Number Theorem. In this paper, we obtain explicit numerical estimates for the average order of its summatory function both in the classical case and in arithmetic progressions. In addition, we derive new explicit estimates for sums over zeros and for the function $ψ(u,χ)$. Our results are general and describe how the explicit bounds depend on other known explicit estimates. These support the known asymptotic results under the (Generalised) Riemann Hypothesis involving error terms.

math.NT↗
arXiv2024

Explicit estimates for the sum $\sum_{k=0}^{n} k! {n\choose k}^2 (-1)^{k}$

We are interested in finding an explicit estimate to the binomial sum $Q_n(x)=\sum_{k=0}^{n} k! {n\choose k}^2 (-x)^{k}$ at $x=1$ for $n=0,1,2,\ldots$. Despite of its own interest the polynomial $Q_n(x)$ is important as the denominator in the Padé identity of the Euler's factorial series $E(x) = \sum_{k=0}^{\infty} k! x^k$ as well as its close connection to a classical Laguerre polynomial $L_n(x) = \frac{1}{n!} e^x \left(\frac{d}{dx}\right)^n (e^{-x}x^n)$. Our main result is the explicit bound $$\left|L_n(1)-\sqrt{\frac{e}π}\cdot \frac{\cos (2\sqrt{n}-\fracπ{4})}{n^{1/4}} +\frac{17}{48}\sqrt{\frac{e}π}\frac{\sin(2\sqrt{n}-\fracπ{4})}{n^{3/4}}\right|<\frac{0.51}{n}$$ for all $n=0,1,2,\ldots$, which replaces the Fejér's asymptotic formula from 1909. As a corollary of this, one also gets a new proof for the bound $|Q_{n}(1)| \le n!$, and even more.

math.NT↗
arXiv2023

Transcendence measure of $e^{1/n}$

For a given transcendental number $ξ$ and for any polynomial $P(X)=: λ_0+\cdots+λ_k X^k \in \mathbb{Z}[X]$, we know that $ P(ξ) \neq 0.$ Let $k \geq 1$ and $ω(k, H)$ be the infimum of the numbers $r > 0$ satisfying the estimate $$ \left|λ_0+λ_1 ξ+λ_2 ξ^{2}+ \ldots +λ_kξ^{k}\right| > \frac{1}{H^r}, $$ for all $(λ_0, \ldots ,λ_k)^T \in \mathbb{Z}^{k+1}\setminus\{\overline{0}\}$ with $\max_{1\le i\le k} \{|λ_i|\} \le H$. Any function greater than or equal to $ω(k, H)$ is a {\it transcendence measure of $ξ$}. In this article, we find out a transcendence measure of $ e^{1/n}$ which improves a result proved by Mahler(\cite{Mahler}) in 1975.

math.NT↗
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↗
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.

math.NT↗
arXiv2017

On the secrecy gain of $\ell$-modular lattices

We show that for every $\ell>1$, there is a counterexample to the $\ell$-modular secrecy function conjecture by Oggier, Solé and Belfiore. These counterexamples all satisfy the modified conjecture by Ernvall-Hytönen and Sethuraman. Furthermore, we provide a method to prove or disprove the modified conjecture for any given $\ell$-modular lattice rationally equivalent to a suitable amount of copies of $\mathbb{Z}\oplus \sqrt{\ell}\,\mathbb{Z}$ with $\ell \in \{3,5,7,11,23\}$. We also provide a variant of the method for strongly $\ell$-modular lattices when $\ell\in \{6,14,15\}$.

cs.IT↗
arXiv2014

Probability Estimates for Fading and Wiretap Channels from Ideal Class Zeta Functions

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, but also on the values of the ideal class Dedekind zeta functions. Along the way, we derive an estimate of the number of elements in a given ideal with a certain algebraic norm within a finite hypercube. We provide several examples which measure the accuracy and predictive ability of our theorems.

cs.IT↗
arXiv2014

Counterexample to the $l$-modular Belfiore-Solé Conjecture

We show that the secrecy function conjecture that states that the maximum of the secrecy function of an $l$-modular lattice occurs at $1/\sqrt{l}$ is false, by proving that the 4-modular lattice $C^(4) = \mathbb{Z} \oplus \sqrt{2}\mathbb{Z} \oplus 2\mathbb{Z}$ fails to satisfy this conjecture. We also indicate how the secrecy function must be modified in the $l$-modular case to have a more reasonable chance for it to have a maximum at $1/\sqrt{l}$, and show that the conjecture, modified with this new secrecy function, is true for various odd 2-modular lattices.

cs.IT↗
arXiv2014

On generalized Li criterion for a certain class of $L-$functions

We define generalized Li coefficients, called $τ-$Li coefficients for a very broad class $\mathcal{S}^{\sharp \flat }(σ_0, σ_1)$ of $L-$functions that contains the Selberg class, the class of all automorphic $L-$functions and the Rankin-Selberg $L-$functions, as well as products of suitable shifts of those functions. We prove the generalized Li criterion for zero-free regions of functions belonging to the class $\mathcal{S}^{\sharp \flat }(σ_0, σ_1)$, derive an arithmetic formula for the computation of $τ-$Li coefficients and conduct numerical investigation of $τ-$Li coefficients for a certain product of shifts of the Riemann zeta function.

math.NT↗
arXiv2013

An explicit Baker type lower bound of exponential values

Let $\mathbb{I}$ denote an imaginary quadratic field or the field $\mathbb{Q}$ of rational numbers and $\mathbb{Z}_{\mathbb{I}}$ its ring of intergers. We shall prove an explicit Baker type lower bound for $\mathbb{Z}_{\mathbb{I}}$-linear form of the numbers \begin{equation}\label{1} 1,\ e^{α_1},...,\ e^{α_m},\quad m\ge 2, \end{equation} where $α_0=0$, $α_1,...,α_m$, are $m+1$ different numbers from the field $\mathbb{I}$. Our work gives gives some improvements to the existing explicit versions of of Baker's work about exponential values at rational points. In particilar, dependences on $m$ are improved.

math.NT↗
arXiv2013

Some results related to the conjecture by Belfiore and Solé

In the first part of the paper, we consider the relation between kissing number and the secrecy gain. We show that on an $n=24m+8k$-dimensional even unimodular lattice, if the shortest vector length is $\geq 2m$, then as the number of vectors of length $2m$ decreases, the secrecy gain increases. We will also prove a similar result on general unimodular lattices. We will also consider the situations with shorter vectors. Furthermore, assuming the conjecture by Belfiore and Solé, we will calculate the difference between inverses of secrecy gains as the number of vectors varies. We will show by an example that there exist two lattices in the same dimension with the same shortest vector length and the same kissing number, but different secrecy gains. Finally, we consider some cases of a question by Elkies by providing an answer for a special class of lattices assuming the conjecture of Belfiore and Solé. We will also get a conditional improvement on some Gaulter's results concerning the conjecture.

cs.IT↗