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Jean-Christophe Pain

Publications and source records attributed to Jean-Christophe Pain.

At least 37 records · Page 2Linked to original sources
arXiv2026

A probabilistic journey through the Newton-Girard identities

This article presents a pedagogical probabilistic exploration of the Newton-Girard identities. We show that the coefficients in these classical relations between power sums and elementary symmetric polynomials can be interpreted as the stable limits of integrals over the unit cube, and as ratios of moments of simple probability distributions. Several classes of integrals are studied, including trigonometric and multiplicative forms. In addition, we discuss the spectral implications via the Le Verrier-Souriau-Faddeev algorithm and Random Matrix Theory, providing a unified framework for the asymptotic algebraic behavior of these identities. While the identities are classical, the probabilistic interpretation of the limits of their normalized forms is the specific focus of the present work.

math.PR↗
arXiv2026

Integral and arithmetic structures of alternating (zigzag) numbers $A_n$

The alternating (zigzag) numbers $A_n$, counting the ascending alternating permutations of $\left\{1,\cdots,n\right\}$ and defined by the exponential generating function $\tan x+\sec x$, admit several classical combinatorial and analytic representations. In this work we unify and extend three complementary structures of $A_n$. First, starting from the Stirling number expansion of zigzag numbers, we derive a contour integral representation, as well as a positive Laplace-type integral representation $$ A_n = 2^n \int_0^\infty e^{-y} f_n(y)\, dy, \qquad f_n(y) := \sum_{k=0}^{n} (-1)^k S(n,k) \left(\frac{y}{2}\right)^k, $$ where the kernel $f_n(y)$ is the polynomial generating function of Stirling numbers. A continuous interpolation of the discrete product (falling factorial) is introduced subsequently. This provides a direct analytic bridge between set partitions and Laplace asymptotics. Second, using the partial fraction expansion of $\tan$, we obtain the well-known hyperbolic integral representation $$ A_{2n+1}=\frac{1}π\int_0^\infty\frac{y^{2n+1}}{\sinh(y/2)}\,dy, $$ equivalently expressed in classical $\cosh$ form for $A_{2n}$. This representation interprets zigzag numbers as spectral moments associated with half-integer poles. The connection with Fourier analysis and Mellin transforms is also outlined. Finally, combining spectral expansions with Stirling identities, we derive congruence relations modulo primes for $A_n$. These results exhibit a dual analytic-combinatorial structure of zigzag numbers, linking partition expansions, trigonometric spectra, and arithmetic properties.

math.CO↗
arXiv2026

Touchard's identity and a detailed determination of the radius of convergence of the Catalan series

While the value of the radius of convergence of the generating series of the Catalan numbers is well-known, obtaining it solely from recurrence relations is less immediate. It is sometimes considered that no known proof establishes that the radius $R$ equals 1/4 without relying on the explicit closed formula for the Catalan numbers. In particular, it has been shown that one can obtain, at the cost of substantial technical effort and without resorting to the main Segner recursion relation or to the explicit formula for the Catalan number, the lower bound $R\geq 1/6$. In this work, we prove that Touchard's recurrence alone yields the optimal exponential upper bound $\limsup_{n\to\infty} C_n^{1/n} \le 4$, which implies $R \ge 1/4$. Combined with the classical lower estimate $\limsup_{n\to\infty} C_n^{1/n} \ge 4$, obtained from central binomial coefficients, this gives $R = 1/4$.

math.CO↗
arXiv2026

Steganographic information hiding via symmetric numerical semigroups

We introduce a steganographic information hiding scheme based on structural properties of numerical semigroups arising from the Frobenius coin problem. Instead of encoding data through representable integers, the proposed protocol embeds information into the gap structure of carefully chosen symmetric numerical semigroups. Symmetry guarantees a balanced gap density, ensuring that encoded values are statistically indistinguishable from uniform numerical noise to an observer lacking the private generating set. The security of the scheme relies on the assumed average-case hardness of numerical semigroup membership inference for hidden generators, offering a novel number-theoretic primitive for covert communication and post-quantum resilient information hiding.

math.CO↗
arXiv2026

Three expressions of the $n$-th prime number: discrete sieving, spectral analysis and probabilistic dynamics

The search for a closed-form expression of the $n$-th prime number, $p_n$, has long oscillated between the rigid determinism of analytic functions and the apparent randomness of local distributions. This paper explores three different approaches to $p_n$. The first one formalizes an analytical identity for $p_{n}$ based on a harmonic summation filtered by a Möbius-derived coprimality indicator. Unlike Gandhi's 1971 identity, which employs a geometric density and logarithmic extraction, this formula operates through a discrete summation over the range defined by Bertrand's postulate. In the second one, we refine the ``harmonic resonance'' model, which posits that primes emerge as spectral nodes from von Mangoldt oscillations. Third, we adopt a ``survival dynamics'' approach, inspired by Mertens' theorems, treating prime spacing as an evolutionary growth process. By bridging these perspectives, we offer a comprehensive framework for understanding the transition from asymptotic trends to discrete arithmetic realities.

math.GM↗
arXiv2026

Generalized relations between arithmetic functions

The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{μ^2(d)}{φ(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right). $$ This formula expresses a non-trivial divisor sum involving the Möbius function $μ$ and Euler's totient function $φ$ as a simple and explicit multiplicative expression. This is a generalization of the remarkable Dineva formula, which corresponds to $s=0$ and gives $n/φ(n)$ on the right-hand side. We explain why only squarefree divisors are involved, show how multiplicativity naturally comes into play, and interpret the identity as a finite Euler product. Beyond this one-parameter family of generalizations, we describe a general method for constructing similar formulas and present several examples. Finally, we reformulate these identities in terms of partial zeta functions, thus emphasizing their close relationship with the classical theory of Euler products and the Riemann zeta function. The connection with the Selberg sieve is briefly outlined.

math.NT↗
arXiv2025

Asymptotic equivalents of partial sums of the reciprocals of prime numbers via the von Mangoldt function

In this paper, we discuss an alternative approach to determine an asymptotic equivalent of the partial sum of the reciprocals of prime numbers. This well-known result, related to Merten's second theorem, is usually derived through methods similar to those found in Hardy and Wright's book ``An introduction to the theory of numbers'', involving comparisons with integrals. The present proof differs in several respects, combining an equivalent for the partial sum of $Λ(m)/m$, where $Λ$ denotes the von Mangoldt function, with an application of Abel's summation formula and properties of the second Chebyshev function $Ψ(x)=\sum_{n\le x}Λ(n)$. A simple application to the study of integers with large prime factors is also presented. Beyond the pedagogical aspect of this work, the aim is to highlight the complementarity of arithmetic functions and to show that interesting (and nontrivial) results can be obtained by means of elementary methods.

math.NT↗
arXiv2025

Excitation and ionization by electron impact in transition and super-transition arrays

This study investigates the ionization and excitation processes induced by electron impact between two configurations or superconfigurations. Rate coefficients are calculated for transition arrays or super-transition arrays rather than level-to-level transitions. Special attention is given to a series of oxygen-like ions relevant to inertial confinement fusion, specifically silicon, germanium, argon, and krypton. Calculations are performed over a wide temperature range, from 50 to 3000 eV. To facilitate the determination of rates at any temperature and ion charge, the computed rates are fitted with a two-dimensional Chebyshev polynomial expansion, yielding a set of coefficients for practical applications. Additionally, an extension of the Clenshaw algorithm to two dimensions, using the Chebyshev coefficients, is proposed to address numerical challenges and enhance computational efficiency.

physics.atom-ph↗
arXiv2025

Modeling stopping power of ions in plasmas using parametric potentials

We present a study of the ion stopping power due to free and bound electrons in a warm dense plasma. Our main goal is to propose a method of stopping-power calculation expected to be valid for any ionization degree. The free-electron contribution is described by the Maynard-Deutsch-Zimmerman formula and the bound-electron one relies on the Bethe formula with corrections, in particular taking into account density and shell effects. The impact of the bound-state computation by three different parametric potentials is investigated within the Garbet formalism for the mean excitation energy. The first parametric potential is due to Green, Sellin and Zachor, the second one was proposed by Yunta, and the third one was introduced by Klapisch in the framework of atomic-structure computations. The results are compared to the ones of self-consistent average-atom calculations. This approach correctly bridges the limits of neutral and fully-ionized matter.

physics.plasm-ph↗
arXiv2025

A sum rule for derangements

We propose a sum rule for derangements. Three different proofs are provided. The first one involves integral representations and the second one relies on the Hermite identity for the integer part of the product of an integer by a real number. The third one, which, unlike the previous ones, is not based on induction, proceeds directly from a recurrence relation for the number of derangements.

math.NT↗
arXiv2025

Statistics of multi-electron states and $J$-levels in atomic configurations

The number and nature of atomic configurations are cornerstones of atomic spectroscopy, especially for the calculation of hot-plasma radiative properties. The knowledge of the distributions of magnetic quantum number $M$ and angular momentum $J$ for $N$ identical fermions in a subshell with half-integer spin $j$ is a prerequisite to the determination of the structure of such configurations. The problem is rather complicated, since the possible occurrence of a specific values of $J$ is governed by the Pauli exclusion principle. Several methods, such as generating functions, recurrence relations or algebraic number theory, for instance via Gaussian polynomials, have proven effective in addressing this issue. However, up to now, no general formula was known. In the present work, we present exact and compact explicit formulas for the number of atomic configurations and for the distributions of the total magnetic quantum number $M$ and angular momentum $J$.

physics.atom-ph↗
arXiv2025

Corrections to radiative rates between atomic configurations

The computation of radiative opacity or emissivity of hot dense matter is a challenging task. It requires accounting for an immense number of energy levels and lines across various excitation and ionization states. Whether in local thermodynamic equilibrium (LTE) or non-LTE plasmas, statistical methods provide significant assistance. Many computational codes are based on the Detailed Configuration Accounting approximation, which involves averaged rates between configurations. In that approach, only the mean energies of the configurations are considered, and the effects of the energy distributions of the levels within the initial and final configurations are typically neglected. A long time ago, Klapisch proposed a method to correct the rates. The corresponding formalism includes the energy shift and variance of the Unresolved Transition Array, as well as the average energies of the configurations. We extend this formalism and investigate its impact on opacity calculations in two specific cases: first, the iron experiment conducted at Sandia National Laboratories under conditions similar to those at the base of the Sun's convective zone, dominated by L-shell 2p-$n$d transitions, and second, laser experiments--still for iron--at much lower temperature. The latter measurements shed light on our understanding of the envelopes of $β$-Cephei-type stars, where the relevant transitions are intra-M-shell $Δn=0$ (3-3) transitions, specifically 3s-3p and 3p-3d, in the XUV range. The issue of ensuring the validity of Kirchhoff's law when plasmas approach LTE is also addressed, and a prescription is proposed, applying both to the standard configuration-to-configuration case and to the aforementioned corrections, which account for the energy distribution of the levels within a configuration.

physics.atom-ph↗
arXiv2025

The Future of Solar modelling: requirements for a new generation of solar models

Helioseismology and solar modelling have enjoyed a golden era thanks to decades-long surveys from ground-based networks such as for example GONG, BiSON, IRIS and the SOHO and SDO space missions which have provided high-quality helioseismic observations that supplemented photometric, gravitational, size and shape, limb-darkening and spectroscopic constraints as well as measurements of neutrino fluxes. However, the success of solar models is also deeply rooted in progress in fundamental physics (equation of state of the solar plasma, high-quality atomic physics computations and opacities, description of convection and the role of macroscopic transport processes of angular momentum and chemicals, such as for example meridional circulation, internal gravity waves, shear-induced turbulence or even convection. In this paper, we briefly outline some key areas of research that deserve particular attention in solar modelling. We discuss the current uncertainties that need to be addressed, how these limit our predictions from solar models and their impact on stellar evolution in general. We outline potential strategies to mitigate them and how multidisciplinary approaches will be needed in the future to tackle them.

astro-ph.SR↗
arXiv2025

Imaginary part of the conductivity using Kramers-Kronig relations

In order to obtain the frequency-dependent photo-absorption in a plasma, both the real and imaginary parts of the AC conductivity are required. The real part can be deduced from the knowledge of the static conductivity (given by the Ziman-Evans formula for instance) and the Drude model. The imaginary part, required for the refraction index, can be obtained using the Kramers-Kronig relations. Usually, it is obtained by complex integration in the complex plane of the usual Kramers-Kronig relations, having $ω'-ω$ in the denominator. However, an alternate form of the Kramers-Kronig relation is often used in physics, especially for determining response functions. It has $ω'^2-ω^2$ in the denominator. We provide two determinations of the imaginary part of the conductivity for this latter form, one using a decomposition into simple elements, and the other involving a complex integration in a quarter of the complex plane.

physics.plasm-ph↗
arXiv2025

Complex pseudo-partition functions in the Configurationally-Resolved Super-Transition-Array approach for radiative opacity

A few years ago, Kurzweil and Hazak developed the Configurationally Resolved Super-Transition-Arrays (CRSTA) method for the computation of hot-plasma radiative opacity. Their approach, based on a temporal integration, is an important refinement of the standard Super-Transition-Arrays (STA) approach, which enables one to recover the underlying structure of the STAs, made of unresolved transition arrays. The CRSTA formalism relies on the use of complex pseudo partition functions, depending on the considered one-electron jump. In this article, we find that, despite the imaginary part, the doubly-recursive relation which was introduced in the original STA method to avoid problems due to alternating-sign terms in partition functions, is still applicable, robust, efficient, and exempt of numerical instabilities. This was rather unexpected, in particular because of the occurrence of trigonometric functions, or Chebyshev polynomials, which can be either positive or negative. We also show that, in the complex case, the recursion relation can be presented in a form where the vector of real and imaginary parts at a given iteration is therefore obtained by a sum of the rotated previous ones.

physics.atom-ph↗
arXiv2025

Numerical modeling of isochoric heating experiments using the TROLL code in the warm dense matter regime

Experiments of isochoric heating by protons of solid material were recently performed at LULI laser facilities. In these experiments, protons, produced from target normal sheath acceleration (TNSA) of Au foil with the PICO2000 laser, deposit their energy into an aluminum or copper foil initially at room temperature and solid density. The heated material reaches the warm dense matter regime with temperature in the rear face of the material between 1 and 5 eV. The temperature is inferred by streaked optical pyrometry and the proton beam is characterized by Thomson parabola. The high-energy protons produced by TNSA are modeled to deduce the initial proton distribution before the slowing down in the target. Hydrodynamic radiative simulations were next performed using the TROLL code in multidimensional geometry. In the TROLL code, the heating of protons is modeled with a Monte-Carlo transport module of charged particle and the calculation of the energy deposited by the protons in the matter is performed using stopping power formulas like SRIM functions. The results of simulations with the TROLL code are compared with the experimental results. An acceptable agreement between experiment and simulation is found for the temperature at the rear of the material using SESAME equation of state and SRIM stopping power for protons in aluminum.

physics.plasm-ph↗
arXiv2025

Explicit formulas for Euler's totient function and the number of divisors

In this article, we present relations for the Euler totient function $φ(n)$ and the number of divisors $τ(n)$ in terms of finite sums of integer parts of rational numbers or greatest common divisors of pairs of integers. Some of the formulas are obtained using a relation due to Menon and the connections with the Pillai arithmetic function are outlined. The reported expressions may be useful to derive new bounds for the usual arithmetical functions.

math.NT↗