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Alexandru Zaharescu

Publications and source records attributed to Alexandru Zaharescu.

At least 55 records · Page 3Linked to original sources
arXiv2024

On the trajectories of a particle in a translation invariant involutive field

We introduce a double-folded operator that, upon iterative application, generates a dynamical system with two types of trajectories: a cyclic one and, another that grows endlessly on parabolas. These trajectories produce two distinct partitions of the set of lattice points in the plane. Our object is to analyze these trajectories and to point out a few special arithmetic properties of the integers they represent. We also introduce and study the parabolic-taxicab distance, which measures the fast traveling on the steps of the stairs defined by points on the parabolic trajectories whose coordinates are based on triangular numbers.

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arXiv2024

A lozenge triangulation of the plane with integers

We introduce and study a three-folded linear operator depending on three parameters that has associated a triangular number tilling of the plane. As a result the set of all triples of integers is decomposed in classes of equivalence organized in four towers of two-dimensional triangulations. We provide the full characterization of the represented integers belonging to each network as families of certain quadratic forms. We note that one of the towers is generated by a germ that produces a covering of the plane with {Löschian} numbers.

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arXiv2024

Evaluations and relations for finite trigonometric sums

Several methods are used to evaluate finite trigonometric sums. In each case, either the sum had not previously been evaluated, or it had been evaluated, but only by analytic means, e.g., by complex analysis or modular transformation formulas. We establish both reciprocity and three sum relations for trigonometric sums. Motivated by certain sums that we have evaluated, we add coprime conditions to the summands and thereby define analogues of Ramanujan sums, which we in turn evaluate. One of these analogues leads to a criterion for the Riemann Hypothesis, analogous to the Franel-Landau criterion.

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arXiv2024

Exponential sums over Möbius convolutions with applications to partitions

We consider partitions $p_{w}(n)$ of a positive integer $n$ arising from the generating functions \[ \sum_{n=1}^\infty p_{w}(n) z^n = \prod_{m \in \mathbb{N}} (1-z^m)^{-w(m)}, \] where the weights $w(m)$ are Möbius convolutions. We establish an upper bound for $p_w(n)$ and, as a consequence, we obtain an asymptotic formula involving the number of odd and even partitions emerging from the weights. In order to achieve the desired bounds on the minor arcs resulting from the Hardy-Littlewood circle method, we establish bounds on exponential sums twisted by Möbius convolutions. Lastly, we provide an explicit formula relating the contributions from the major arcs with a sum over the zeros of the Riemann zeta-function.

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arXiv2023

Partitions into semiprimes

Let $\mathbb{P}$ denote the set of primes and $\mathcal{N}\subset \mathbb{N}$ be a set with arbitrary weights attached to its elements. Set $\mathfrak{p}_{\mathcal{N}}(n)$ to be the restricted partition function which counts partitions of $n$ with all its parts lying in $\mathcal{N}$. By employing a suitable variation of the Hardy-Littlewood circle method we provide the asymptotic formula of $\mathfrak{p}_{\mathcal{N}}(n)$ for the set of semiprimes $\mathcal{N} = \{p_1 p_2 : p_1, p_2 \in \mathbb{P}\}$ in different set-ups (counting factors, repeating the count of factors, and different factors). In order to deal with the minor arc, we investigate a double Weyl sum over prime products and find its corresponding bound thereby extending some of the results of Vinogradov on partitions. We also describe a methodology to find the asymptotic partition $\mathfrak{p}_{\mathcal{N}}(n)$ for general weighted sets $\mathcal{N}$ by assigning different strategies for the major, non-principal major, and minor arcs. Our result is contextualized alongside other recent results in partition asymptotics.

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arXiv2023

A visual perspective on the Birch and Swinnerton-Dyer conjecture through a family of approximations of $L$-functions

We investigate the properties of a family of approximations of the Hasse-Weil $L$-function associated to an elliptic curve $E$ over $\mathbb{Q}$. We give a precise expression for the error of the approximations, and provide a visual interpretation of the analytic rank $m$ of $E$ as a sequence of near regular polygons around the center of the critical strip, each with vertices at the zeros of the approximations.

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arXiv2023

The Riemann zeta function and exact exponential sum identities of divisor functions

We prove an explicit integral formula for computing the product of two shifted Riemann zeta functions everywhere in the complex plane. We show that this formula implies the existence of infinite families of exact exponential sum identities involving the divisor functions, and we provide examples of these identities. We conjecturally propose a method to compute divisor functions by matrix inversion, without employing arithmetic techniques.

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arXiv2023

Counterintuitive patterns on angles and distances between lattice points in high dimensional hypercubes

Let $\mathcal{S}$ be a finite set of integer points in $\mathbb{R}^d$, which we assume has many symmetries, and let $P\in\mathbb{R}^d$ be a fixed point. We calculate the distances from $P$ to the points in $\mathcal{S}$ and compare the results. In some of the most common cases, we find that they lead to unexpected conclusions if the dimension is sufficiently large. For example, if $\mathcal{S}$ is the set of vertices of a hypercube in $\mathbb{R}^d$ and $P$ is any point inside, then almost all triangles $PAB$ with $A,B\in\mathcal{S}$ are almost equilateral. Or, if $P$ is close to the center of the cube, then almost all triangles $PAB$ with $A\in \mathcal{S}$ and $B$ anywhere in the hypercube are almost right triangles.

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arXiv2023

Distribution of angles to lattice points seen from a fast moving observer

We consider a square expanding with constant speed seen from an observer moving away with constant acceleration and study the distribution of angles between rays from the observer towards the lattice points in the square. We prove the existence of the gap distribution as time tends to infinity and provide explicit formulas for the corresponding density function.

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arXiv2023

On Quasi-Periodicity in Proth-Gilbreath Triangles

Let PG be the Proth-Gilbreath operator that transforms a sequence of integers into the sequence of the absolute values of the differences between all pairs of neighbor terms. Consider the infinite tables obtained by successive iterations of PG applied to different initial sequences of integers. We study these tables of higher order differences and characterize those that have near-periodic features. As a biproduct, we also obtain two results on a class of formal power series over the field with two elements F2 that can be expressed as rational functions in several ways.

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arXiv2023

Proofs of McIntosh's Conjecture on Franel Integrals and Two Generalizations

We provide a proof of a conjecture made by Richard McIntosh in 1996 on the values of the Franel integrals, $$\int_0^1((ax))((bx))((cx))((ex))\,dx,$$ where $((x))$ is the first periodic Bernoulli function. Secondly, we extend our ideas to prove a similar theorem for $$\int_0^1((a_1x))((a_2x))\cdots ((a_{n}x))\,dx.$$ Lastly, we prove a further generalization in which $((x))$ is replaced by any particular Bernoulli function with odd index.

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arXiv2023

Modified Bessel Functions in Analytic Number Theory

The modified Bessel functions $K_ν(z)$, or, for brevity, K-Bessel functions, arise at key places in analytic number theory. In particular, they appear in beautiful arithmetic identities. A survey of these arithmetical identities and their appearances in number theory is provided.

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arXiv2022

A problem of Erdős-Graham-Granville-Selfridge on integral points on hyperelliptic curves

Erdős, Graham, and Selfridge considered, for each positive integer $n$, the least value of $t_n$ so that the integers $n+1, n+2, \dots, n+t_n $ contain a subset the product of whose members with $n$ is a square. An open problem posed by Granville concerns the size of $t_n$, under the assumption of the ABC Conjecture. We establish some results on the distribution of $t_n$, and in the process solve Granville's problem unconditionally.

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arXiv2022

Exact evaluations and reciprocity theorems for finite trigonometric sums

We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of the authors. Reciprocity theorems for certain trigonometric sums are also established.

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arXiv2022

A Generalization of the Schur-Siegel-Smyth Trace Problem

Let $α$ be a totally positive algebraic integer, and define its absolute trace to be $\frac{Tr(α)}{\text{deg}(α)}$, the trace of $α$ divided by the degree of $α$. Elementary considerations show that the absolute trace is always at least one, while it is plausible that for any $ε>0$, the absolute trace is at least $2-ε$ with only finitely many exceptions. This is known as the Schur-Siegel-Smyth trace problem. Our aim in this paper is to show that the Schur-Siegel-Smyth trace problem can be considered as a special case of a more general problem.

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arXiv2022

Ramification of $p$-power torsion points of formal groups

Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of $\mathbb{Q}_p$, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and let $\overline{K}$ be some fixed algebraic closure of $K$. Let $A$ be an abelian variety defined over $F$, with good reduction, let $\mathcal{A}$ denote the Néron model of $A$ over ${\rm Spec}(\mathcal{O}_F)$, and let $\widehat{\mathcal{A}}$ be the formal completion of $\mathcal{A}$ along the identity of its special fiber, i.e. the formal group of $A$. In this work, we prove two results concerning the ramification of $p$-power torsion points on $\widehat{\mathcal{A}}$. One of our main results describes conditions on $\widehat{\mathcal{A}}$, base changed to $\text{Spf}(\mathcal{O}_K) $, for which the field $K(\widehat{\mathcal{A}}[p])/K$ is a tamely ramified extension where $\widehat{\mathcal{A}}[p]$ denotes the group of $p$-torsion points of $\widehat{\mathcal{A}}$ over $\mathcal{O}_{\overline{K}}$. This result generalizes previous work when $A$ is $1$-dimensional and work of Arias-de-Reyna when $A$ is the Jacobian of certain genus 2 hyperelliptic curves.

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