Searcharxiv⌕ Search

arXiv subjects

Es-said En-naoui

Publications and source records attributed to Es-said En-naoui.

9 recordsLinked to original sources
arXiv2025

The Exponential Congruence Symbol

In this work, we study the generalized k-th power symbol (a/n)_k and present a comprehensive collection of its algebraic properties. The results are classified according to their dependence on the three main parameters a, n, and k. In particular, we discuss multiplicativity, inversion, power compatibility, and invariance modulo n with respect to the parameter a. For n, we examine factorization properties, behavior on prime powers, orthogonality relations, and Kummer splitting criteria. Regarding k, we include specialization to classical symbols, k-th reciprocity laws, relations between orders, and embedding into roots of unity. Moreover, we extend the existing theory by providing new essential results, including additive behavior under characters, Mobius filtering, compatibility with Carmichael and Euler functions, and connections with Dirichlet L-series. Finally, we analyze the case where a, n, and k are primes and present mixed results that generalize classical reciprocity laws, Frobenius automorphisms, and Sato-Tate distributions. These results unify and extend previous studies on k-th power symbols and offer a foundation for further arithmetic, algebraic, and analytic investigations.

math.GM↗
arXiv2025

The Theory of Topo-Symmetric Extensions of Topological Groups

We introduce the notion of \emph{topo-symmetric extensions} of topological groups, a new generalization of classical group extensions that incorporates both topological and symmetry constraints. We define morphisms between such extensions, construct the associated groupoid, and develop classification results in terms of adapted cohomology. Several new invariants are introduced, including dimension, stabilizer, and density invariants, which characterize the fine structure of these extensions. Applications are given for finite groups, compact Lie groups, and profinite groups. This theory extends classical cohomological correspondence theorems \cite{MacLane1963, EilenbergMacLane1947}, while opening new perspectives in arithmetic, asymptotic distribution, and congruence properties \cite{HardyWright2008, Serre1979}. Finally, we propose conjectures and open problems concerning density, maximal orders, and modular distribution of topo-symmetric extensions.

math.GM↗
arXiv2025

Extended Mittag--Leffler Stability Results for Fuzzy Fractional Systems

We extend Lyapunov--type Mittag--Leffler stability analysis for fuzzy nonlinear fractional differential equations (Caputo sense) and introduce a family of stronger, broadly applicable stability results. In particular, we develop (i) uniform Mittag--Leffler stability for variable-order Caputo derivatives, (ii) input-to-state stability (ISS) in the Mittag--Leffler sense and explicit robustness/ultimate bound estimates, (iii) a converse Mittag--Leffler Lyapunov theorem, (iv) a fractional LaSalle invariance principle adapted to the fuzzy setting, and (v) practical and computational criteria including Lyapunov--Krasovskii functionals for delay systems and levelwise LMI tests for linearized models. The paper presents precise assumptions, proof sketches, and preparatory lemmas in the preliminaries; subsequent sections (Step 2 onward) provide rigorous proofs and illustrative examples.

math.GM↗
arXiv2025

A Universal Space of Arithmetic Functions:The Banach--Hilbert Hybrid Space U

We introduce a new functional space U designed to contain all classical arithmetic functions (Mobius, von Mangoldt, Euler phi, divisor functions, Dirichlet characters, etc.). The norm of U combines a Hilbert-type component, based on square summability of Dirichlet coefficients for every s > 1, with a Banach component controlling logarithmic averages of partial sums. We prove that U is a complete Banach space which embeds continuously all standard Hilbert spaces of Dirichlet series and allows natural actions of Dirichlet convolution and shift operators. This framework provides a unified analytic setting for classical and modern problems in multiplicative number theory.

math.GM↗
arXiv2025

The Modular Energy Function: Exact Identities and a Primality Equivalence

We introduce and study the arithmetic function E_m(n), defined as the sum of the remainders of n when divided by the first m positive integers. Although the definition is elementary, the function encodes rich arithmetic structure. In this first part we develop exact algebraic and combinatorial identities, including floor-sum and divisor-sum reformulations, quotient-residue decompositions, sharp bounds and extremal classes, and symmetry relations. We also establish a diagonal identity linking E_n(n) to the classical sum of divisors function, which leads to an exact equivalence characterizing primality in terms of differences of modular energy values. Additional results include congruence decompositions, recursion formulas, and structural properties when n m, or when n or m is prime. These identities form the foundation for a second part, where analytic and asymptotic properties will be investigated.

math.GM↗
arXiv2025

Arithmetic and Asymptotic Properties of Restricted Totient Sums

This article extends our previous study on the summatory behavior of Euler's totient function $φ(n)$. We investigate two complementary restricted sums, $Υ(x,p)=\sum_{\substack{k\le x\\\gcd(k,p)=1}}φ(k)$ and $Δ(x,p)=\sum_{\substack{k\le x\\p\mid k}}φ(k)$, which satisfy the decomposition $Ψ(x)=\sum_{k\le x}φ(k)=Υ(x,p)+Δ(x,p)$. We establish recurrence formulas, congruence relations, and generating function identities for $Δ(x,p)$. In particular, we prove that $Δ(x,p)\equiv 0\pmod{p-1}$ for every prime $p$, and we derive the asymptotic expansion $Δ(x,p)=\dfrac{3}{π^{2}(p+1)}\,x^{2}+O(x\log x)$. Furthermore, we study average orders, connections with $ω(n)$, and relations with divisor structures. These results refine the analytic understanding of totients in arithmetic progressions and complement the classical asymptotic theory of $Ψ(x)$.

math.GM↗
arXiv2022

The Dirichlet series of the arithmetic derivative

The main object of this paper is to give the generalized von mangoldt function using the L-additive function which can help us to make it possible to calculate The Dirichlet series of the arithmetic derivative $δ$ and Dirichlet series defined by: $$\sum \limits_{n\geq 1} \frac{f(n)δ(n)}{n^s}$$ where $f$ is a classical arithmetic function.

math.GM↗