arXiv subjects

Sébastien Darses

Publications and source records attributed to Sébastien Darses.

17 recordsLinked to original sources

arXiv2026

Weighted mean squares of $ζ$ and $L(\cdot,χ_4)$: closed forms, reciprocity, and a missing identity

This short note is a concise yet complete overview, intended for students, researchers, and AIs, bringing together in a unified notation the closed-form and reciprocity identities arising from weighted mean squares of the Riemann zeta function and Dirichlet $L$-functions on the critical line. In addition, we prove a new reciprocity identity: the exponentially weighted counterpart of a formula of Lewis and Zagier for the primitive character of modulus $4$.

math.NT↗

arXiv2023

Multiple integral formulas for weighted zeta moments: the case of the sixth moment

We prove exact formulas for weighted $2k$th moments of the Riemann zeta function for all integer $k\geq 1$ in terms of the analytic continuation of an auto-correlation function. This latter enjoys several functional equations. One of them, following from a fundamental lemma of Bettin and Conrey (2013), yields to new formulas for the moments: our second main result is the case $k=3$, but there is no obstruction to obtain higher moments. This generalizes results by Titchmarsh (1928) for $k=1$ and $k=2$. A basic and powerful tool is a special Fourier transform unveiled by Ramanujan (1915). In a nutshell, the new idea is to consider the associated structures to $Γ^kζ^k$, which enjoy remarkable properties that are not satisfied by the more classically studied structure $Γζ^k$.

math.NT↗

arXiv2023

Polynomial Moments with a weighted Zeta Square measure on the critical line

We prove closed-form identities for the sequence of moments $\int t^{2n}|Γ(s)ζ(s)|^2dt$ on the whole critical line $s=1/2+it$. They are finite sums involving binomial coefficients, Bernoulli numbers, Stirling numbers and $π$, especially featuring the numbers $ζ(n)B_n/n$ unveiled by Bettin and Conrey. Their main power series identity, together with our previous work, allows for a short proof of an auxiliary result: the computation of the $k$-th derivatives at $1$ of the "exponential auto-correlation" function studied in \cite{DH21a}. We also provide an elementary and self-contained proof of this secondary result. The starting point of our work is a remarkable identity proven by Ramanujan in 1915. %today interpreted as a Mellin-Plancherel isometry involving the $Γ$ and $ζ$ functions. The sequence of moments studied here, not to be confused with the moments of the Riemann zeta function, entirely characterizes $|ζ|$ on the critical line. They arise in some generalizations of the Nyman-Beurling criterion, but might be of independent interest for %various other applications, as well as for the numerous connections concerning the above mentioned numbers.

math.NT↗

arXiv2022

Polynomial approximations in a generalized Nyman-Beurling criterion

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $θ_k\in(0,1)$, $k\ge1$. Randomizing the $θ_k$ generates new structures and criteria. One of them is a sufficient condition for RH that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper generalizes conditions (i) and (ii) that involve a $σ_0\in(1/2,1)$, and imply $ζ(σ+it)\neq 0$ in the strip $1/2<σ\leσ_0<1$. We then identify functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In this context, the difficulty for proving RH is then reallocated in (ii), which heavily relies on the corresponding Gram matrices, for which two remarkable structures are obtained. We show that a particular tuning of the approximating sequence leads to a striking simplification of the second Gram matrix, then reading as a block Hankel form.

math.FA↗

arXiv2021

On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function

The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on $(0,\infty)$, which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors $θ_k\in(0,1)$, $k\ge1$. We develop probabilistic extensions of the Nyman-Beurling criterion by considering these $θ_k$ as random: this yields new structures and criteria, one of them having a significant overlap with the general strong Báez-Duarte criterion. We start here the study of these criteria, with a special focus on exponential and gamma distributions. The main goal of the present paper is the study of the interplay between these probabilistic Nyman-Beurling criteria and the Riemann hypothesis. We are able to obtain equivalences in two main classes of examples: dilated structures as exponential $\cal E(k)$ distributions, and random variables $Z_{k,n}$, $1\le k\le n$, concentrated around $1/k$ as $n$ is growing. By means of our probabilistic point of view, we bring an answer to a question raised by Báez-Duarte in 2005: the price to pay to consider non compactly supported kernels is a controlled condition on the coefficients of the involved approximations.

math.NT↗

arXiv2020

An exponentially averaged Vasyunin formula

We prove a Vasyunin-type formula for an autocorrelation function arising from a Nyman-Beurling criterion generalized to a probabilistic framework. This formula can also be seen as a reciprocity formula for cotangent sums, related to the ones proven in [BC13], [ABB17].

math.NT↗

arXiv2012

Sparse recovery with unknown variance: a LASSO-type approach

We address the issue of estimating the regression vector $β$ in the generic $s$-sparse linear model $y = Xβ+z$, with $β\in\R^{p}$, $y\in\R^{n}$, $z\sim\mathcal N(0,\sg^2 I)$ and $p> n$ when the variance $\sg^{2}$ is unknown. We study two LASSO-type methods that jointly estimate $β$ and the variance. These estimators are minimizers of the $\ell_1$ penalized least-squares functional, where the relaxation parameter is tuned according to two different strategies. In the first strategy, the relaxation parameter is of the order $\chσ \sqrt{\log p}$, where $\chσ^2$ is the empirical variance. %The resulting optimization problem can be solved by running only a few successive LASSO instances with %recursive updating of the relaxation parameter. In the second strategy, the relaxation parameter is chosen so as to enforce a trade-off between the fidelity and the penalty terms at optimality. For both estimators, our assumptions are similar to the ones proposed by Candès and Plan in {\it Ann. Stat. (2009)}, for the case where $\sg^{2}$ is known. We prove that our estimators ensure exact recovery of the support and sign pattern of $β$ with high probability. We present simulations results showing that the first estimator enjoys nearly the same performances in practice as the standard LASSO (known variance case) for a wide range of the signal to noise ratio. Our second estimator is shown to outperform both in terms of false detection, when the signal to noise ratio is low.

math.ST↗

arXiv2012

Invertibility of random submatrices via tail decoupling and a Matrix Chernoff Inequality

Let $X$ be a $n\times p$ matrix with coherence $μ(X)=\max_{j\neq j'} |X_j^tX_{j'}|$. We present a simplified and improved study of the quasi-isometry property for most submatrices of $X$ obtained by uniform column sampling. Our results depend on $μ(X)$, $\|X\|$ and the dimensions with explicit constants, which improve the previously known values by a large factor. The analysis relies on a tail decoupling argument, of independent interest, and a recent version of the Non-Commutative Chernoff inequality (NCCI).

math.PR↗

arXiv2010

Limit theorems for nonlinear functionals of Volterra processes via white noise analysis

By means of white noise analysis, we prove some limit theorems for nonlinear functionals of a given Volterra process. In particular, our results apply to fractional Brownian motion (fBm) and should be compared with the classical convergence results of the 1980s due to Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu, as well as the recent advances concerning the construction of a Lévy area for fBm due to Coutin, Qian and Unterberger.

math.PR↗

arXiv2008

Lagrangian structures for the Stokes, Navier-Stokes and Euler equations

We prove that the Navier-Stokes, the Euler and the Stokes equations admit a Lagrangian structure using the stochastic embedding of Lagrangian systems. These equations coincide with extremals of an explicit stochastic Lagrangian functional, i.e. they are stochastic Lagrangian systems in the sense of [Cresson-Darses, J. Math. Phys. 48, 072703 (2007]

math.AP↗

arXiv2008

Asymptotic expansions at any time for scalar fractional SDEs with Hurst index $H>1/2$

We study the asymptotic expansions with respect to $h$ of \[\mathrm{E}[Δ_hf(X_t)],\qquad \mathrm{E}[Δ_hf(X_t)|\mathscr{F}^X_t]\quadand\quad \mathrm{E}[Δ_hf(X_t)|X_t],\] where $Δ_hf(X_t)=f(X_{t+h})-f(X_t)$, when $f:\mathbb {R}\to\mathbb{R}$ is a smooth real function, $t\geq0$ is a fixed time, $X$ is the solution of a one-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst index $H>{1}/{2}$ and $\mathscr{F}^X$ is its natural filtration.

math.PR↗

arXiv2007

Stochastic derivatives for fractional diffusions

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given $σ$-field $\mathcal{Q}$. In our framework, we recall well-known results about Markov--Wiener diffusions. We then focus mainly on the case where $X$ is a fractional diffusion and where $\mathcal{Q}$ is the past, the future or the present of $X$. We treat some crucial examples and our main result is the existence of stochastic derivatives with respect to the present of $X$ when $X$ solves a stochastic differential equation driven by a fractional Brownian motion with Hurst index $H>1/2$. We give explicit formulas.

math.PR↗

arXiv2007

Differentiating sigma-fields for Gaussian and shifted Gaussian processes

We study the notions of differentiating and non-differentiating sigma-fields in the general framework of (possibly drifted) Gaussian processes, and characterize their invariance properties under equivalent changes of probability measure. As an application, we investigate the class of stochastic derivatives associated with shifted fractional Brownian motions. We finally establish conditions for the existence of a jointly measurable version of the differentiated process, and we outline a general framework for stochastic embedded equations.

math.PR↗

arXiv2006

Dynamical properties and characterization of gradient drift diffusions

We study the dynamical properties of the Brownian diffusions having $σ{\rm Id}$ as diffusion coefficient matrix and $b=\nabla U$ as drift vector. We characterize this class through the equality $D^2_+=D^2_-$, where $D_{+}$ (resp. $D_-$) denotes the forward (resp. backward) stochastic derivative of Nelson's type. Our proof is based on a remarkable identity for $D_+^2-D_-^2$ and on the use of the martingale problem. We also give a new formulation of a famous theorem of Kolmogorov concerning reversible diffusions. We finally relate our characterization to some questions about the complex stochastic embedding of the Newton equation which initially motivated of this work.

math.PR↗

arXiv2005

Plongement stochastique des systèmes lagrangiens

We define an operator which extends classical differentiation from smooth deterministic functions to certain stochastic processes. Based on this operator, we define a procedure which associates a stochastic analog to standard differential operators and ordinary differential equations. We call this procedure stochastic embedding. By embedding lagrangian systems, we obtain a stochastic Euler-Lagrange equation which, in the case of natural lagrangian systems, is called the embedded Newton equation. This equation contains the stochastic Newton equation introduced by Nelson in his dynamical theory of brownian diffusions. Finally, we consider a diffusion with a gradient drift, a constant diffusion coefficient and having a probability density function. We prove that a necessary condition for this diffusion to solve the embedded Newton equation is that its density be the square of the modulus of a wave function solution of a linear Schrödinger equation.

math.PR↗

arXiv2005

Lemme de coherence et théorème de Noether stochastique

The stochastic embedding procedure associates a stochastic Euler-Lagrange equation (SEL) to the standard Euler-Lagrange equation (EL). Can we derive (SEL) from a generalized least action principle? To address this question, we develop a stochastic calculus of variation initiated by Yasue. We give a stochastic analog F of the lagrangian action functional. We introduce a notion of stationarity according to which the solutions of (SEL) are the stationary points of F. This notion of stationarity brings coherence to stochastic calculus of variation with respect to stochastic embedding. Finally, we prove a stochastic Noether theorem which introduces an original notion of stochastic first integral.

math.PR↗

arXiv2005

Stochastic embedding of dynamical systems

Most physical systems are modelled by an ordinary or a partial differential equation, like the n-body problem in celestial mechanics. In some cases, for example when studying the long term behaviour of the solar system or for complex systems, there exist elements which can influence the dynamics of the system which are not well modelled or even known. One way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects, that are eventually stochastic. In this paper, we develop a theory for the stochastic embedding of ordinary differential equations. We apply this method to Lagrangian systems. In this particular case, we extend many results of classical mechanics namely, the least action principle, the Euler-Lagrange equations, and Noether's theorem. We also obtain a Hamiltonian formulation for our stochastic Lagrangian systems. Many applications are discussed at the end of the paper.

math.PR↗