arXiv subjects
Publications and source records attributed to Walid Oukil.
We introduce the quantity $μ_η$, defined for every complex $s$ in the critical strip, as a transformation of the Mellin transform associated to the functions $η$. We establish a sufficient condition on $η$ under which $μ_η(s)$ and $μ_η(1-s)$ cannot both vanish outside the critical line. An application is given to the case in which $η$ is the fractional part function, and the zeros of $μ_η$ coincide with the zeros of the Riemann zeta function.
We propose a unified framework for Prime Rigidity Theory (PR) by introducing the notions of an "abstract integer system" and an "abstract prime set", consisting, respectively, of two unbounded sequences $\mathcal{P}$ and $\mathcal{N}$ of elements of $\mathcal{X}$, where $\mathcal{X}$ denotes the algebra of bounded linear endomorphisms of a complex Banach space $X$, such that every element of $\mathcal{N}$ can be expressed as a finite product of powers of elements of $\mathcal{P}$. We define the zeta function $ζ_{X,\mathcal{N}}$ on $\mathcal{X}\setminus{I_X}$, where $I_X$ is the identity endomorphism. The scalar restriction $ζ_{\mathbb{C},\mathcal{N}}$ yields generalized Beurling zeta functions; in particular, $ζ_{\mathbb{C},\mathbb{N}}$ coincides with the classical Riemann zeta function. We introduce the notions of a "rigid abstract integer system" and "rigid abstract prime set" in order to preserve the rigid asymmetric structure obtained in the scalar case.
The present note identifies the fundamental mechanism governing the extension of the contraction method. Although every periodic vector field defined by a finite trigonometric polynomial admits a bounded deviation from a linear drift, this property fails in general for smooth periodic vector fields. We show that the contraction argument underlying the original proof extends to any field satisfying a natural uniform summability condition, and that the counterexample violates this condition, thereby revealing the obstruction that prevents the method from extending beyond the finite-spectrum setting. For a smooth periodic vector field on the $n$-torus, the contraction method for establishing strong rotation vectors extends only to those asymptotic directions $ρ\in \mathbb{R}^n$ for which a certain spectral sum remains uniformly bounded along a sequence of rational approximations. We introduce the "arithmetic cone" $\mathfrak{C}(f)$, defined as the set of all $ρ$ admitting such an approximation. We establish its basic algebraic property: it is a cone. We prove that, under a uniform contraction condition, every element of $\mathfrak{C}(f)$ yields a strong rotation vector for the dynamics. The construction reveals a precise link between the Fourier asymptotics of $f$ and the arithmetic of admissible rotation directions. In the second part, we introduce the class of "spectrally admissible" fields $\mathcal{A}_{spec}$, for which the cone of the augmented field equals the whole space, and we show that it contains all finite trigonometric polynomials.
In this manuscript, we construct an explicit counterexample of smooth infinitely differentiable, periodic dynamical system on the 3-torus for which the rotation vector exists in the weak sense but fails to exist in the strong sense of bounded deviation. The construction uses Liouville-type arithmetic resonances and shows that periodicity and infinite differentiability alone do not guarantee a bounded drift deviation, even for integrable flows. To the best of our knowledge, this is the first example of a smooth integrable torus flow with unbounded rotational deviation whose Jacobian is everywhere nilpotent.: the Jacobian matrix is strictly nilpotent, all its eigenvalues are identically zero, and consequently no local exponential stretching or contraction occurs anywhere. The unbounded deviation from the linear drift is generated purely by the accumulation of infinitely many incommensurable frequencies, without any amplification mechanism. This purely neutral local dynamics makes the example particularly relevant for coupled phase oscillator models, where the linearized dynamics around a synchronized state is typically nilpotent or neutral. Moreover, this work extends the theory of trigonometric polynomial fields. It was previously shown that when the frequency spectrum $Λ_f$ is finite, a strong rotation vector always exists. The present counterexample demonstrates that as soon as the spectrum becomes infinite (while retaining full regularity) the strong rotation vector can disappear. Thus, the finiteness of the spectrum cannot be replaced by smoothness alone.
In this manuscript, we consider the Riemann zeta function $ζ$, defined through the Abel summation formula. We present a simple analytical method based on a complex differential equation. The aim is to propose a new analytical approach, relying on complex differential equations defined on the interval $[1,+\infty)$, in order to gain insight into the behavior of $ζ(s)$ within the critical strip. We introduce a differential equation depending only on the complex parameter $s$, extracted from the analytical structure of $ζ(s)$ for $s$ in the critical strip. This equation admits a unique continuous and bounded solution. The non-trivial zeros of the zeta function can thus be characterized through the boundedness of such a solution. Furthermore, we conjecture an asymmetry in the boundedness of these solutions with respect to the critical line, suggesting that if $ζ(1-s)= 0$, then $ζ(s) \neq 0$ for any $s$ in the critical strip except on the critical line. This observation does not contradict the Riemann functional equation but supports a formulation consistent with the Riemann Hypothesis, opening a simple yet potentially new direction for the analytical investigation of the zeta function and the localization of its non-trivial zeros.
This paper investigates the exponential stability of abstract mean field systems in their synchronized state. We analyze stability by studying the linearized system and demonstrate the existence of an exponentially stable invariant manifold. Our focus is on the equilibrium stability under synchronization. We provide a comprehensive analysis of both linear and nonlinear cases of the system. Additionally, we prove the existence of stable limit cycles and establish a relation between the dynamics in linear and nonlinear frameworks.