相关论文: Moments of restricted divisor functions

We investigate the first and second moments of shifted convolutions of the generalised divisor function $d_3(n)$.

We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta-function averaged over a subfamily of zeros of the zeta function which is expected to have full density inside the set of all zeros. For…

We study higher moments of convolutions of the characteristic function of a set, which generalize a classical notion of the additive energy. Such quantities appear in many problems of additive combinatorics as well as in number theory. In…

The multipolar decomposition of current distributions is used in many branches of physics. Here, we obtain new exact expressions for the dipolar moments of a localized electric current distribution. The typical integrals for the dipole…

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine…

This paper is devoted to hydrodynamic limits of linear kinetic equations when the thermodynamical equilibrium is described by a heavy-tail distribution function rather than a Maxwellian distribution. We show that the long time/small mean…

Let $f$ be a Steinhaus random multiplicative function, and for $\alpha\in \mathbb{R}$, let $d_\alpha$ denote the $\alpha$-divisor function. For $\alpha \in (1,2)$ we establish that $$ \mathbb{E}\bigg\{\Big|\frac{1}{\sqrt{x}}\sum_{n\leq x}…

Distributed-order time-fractional wave equations appear in the modeling of wave propagation in viscoelastic media. The material characteristics of the medium are modeled through constitutive functions or distributions in the…

We study the moments of the Dirichlet L-function when defined over the polynomial ring over finite fields. We find an asymptotic formula to the fourth moment of the central value of Dirichlet L functions in this context. We also find a…

The discrete moment problem is a foundational problem in distribution-free robust optimization, where the goal is to find a worst-case distribution that satisfies a given set of moments. This paper studies the discrete moment problems with…

Ratios of random variables often appear in probability and statistical applications. We aim to approximate the moments of such ratios under several dependence assumptions. Extending the ideas in Collomb [C. R. Acad. Sci. Paris 285 (1977)…

In this note, we derive the closed form formulae for moments of Student's t-distribution in the one dimensional case as well as in higher dimensions through a unified probability framework. Interestingly, the closed form expressions for the…

We derive a moment formula for generalized fractional polynomial processes, i.e., for polynomial-preserving Markov processes time-changed by an inverse L\'evy-subordinator. If the time change is inverse $\alpha$-stable, the time-derivative…

This paper describes a method to compute lower bounds for moments of $\zeta$ and $L$-functions. The method is illustrated in the case of moments of $|\zeta(\frac 12+it)|$, where the results are new for small moments $0< k<1$.

The paper studies generalized differentiability properties of the marginal function of parametric optimal control problems of semilinear elliptic partial differential equations. We establish upper estimates for the regular and the limiting…

Assuming the Generalized Riemann Hypothesis and the Generalized Ramanujan Conjecture, we determine the order of the $2(k_1,\dots,k_r)$th moment of a product of distinct irreducible $L$-functions on the critical line. As a consequence, we…

We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower…

The uniqueness question of the multivariate moment problem is studied by different methods: Hilbert space operators, complex function theory, polynomial approximation, disintegration, integral geometry. Most of the known results in the…

We consider diffusion type equations with a distributed order derivative in the time variable. This derivative is defined as the integral in $\alpha$ of the Caputo-Dzhrbashian fractional derivative of order $\alpha \in (0,1)$ with a certain…

We propose threshold diffusion processes as unique solutions to stochastic differential equations with step-function coefficients, and obtain explicit expressions for the conditional Laplace transform of the hitting times and the potential…