Related papers: Approximating the divisor functions

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents…

Let $d(n)$ be the Dirichlet divisor function and $\Delta(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$. In this paper we focus on the sum $\sum_{p\leqslant x}\Delta^2(p)$, where $p$ runs over…

We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which…

The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for…

Previously, the author offered a plasma-like description of quantum phenomena. This article offers a new criterion of approximation of probability density functions of quantum theories by sums of $\delta$-functions with integer coefficients…

We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known…

We study the asymptotic behavior of the sums of divisors when the integers are modelled with the Bernoulli random walk; We prealably study the correlation properties of the corresponding system.

We introduce a new class of multiplications of distributions in one dimension merging together two different regularizations of distributions. Some of the features of these multiplications are discussed in a certain detail. We use our…

The functional delta-method provides a convenient tool for deriving the asymptotic distribution of a plug-in estimator of a statistical functional from the asymptotic distribution of the respective empirical process. Moreover, it provides a…

We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture…

We obtain asymptotic approximations for the probability density function of the product of two correlated normal random variables with non-zero means and arbitrary variances. As a consequence, we deduce asymptotic approximations for the…

We consider a sum of the derivatives of Dirichlet $L$-functions over the zeros of Dirichlet $L$-functions. We give an asymptotic formula for the sum.

In this paper, we present an improved continued fraction approximation of the Wallis ratio. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the double-side inequality related to…

We construct $\Delta$-operators $F[\Delta]$ on the space of almost symmetric functions $\mathscr{P}_{as}^{+}$. These operators extend the usual $\Delta$-operators on the space of symmetric functions $\Lambda \subset \mathscr{P}_{as}^{+}$…

We prove new exact formulas for the generalized sum-of-divisors functions, $\sigma_{\alpha}(x) := \sum_{d|x} d^{\alpha}$. The formulas for $\sigma_{\alpha}(x)$ when $\alpha \in \mathbb{C}$ is fixed and $x \geq 1$ involves a finite sum over…

We study the approximation of functions which are invariant with respect to certain permutations of the input indices using flow maps of dynamical systems. Such invariant functions includes the much studied translation-invariant ones…

We study the Cowling approximation by analytical means as applied to a system of linear differential equations arising from models of non-radial stellar pulsation. We consider various asymptotic cases, including those of high harmonic…

It is shown how the linear method of the Yosida-approximation of the derivative applies to solve possibly nonlinear abstract functional differential equations in both, the finite and infinite delay case. A generalization of the integral…

A new continuity for set-valued functions is introduced, and an existence theorem is proved for such continuous set-valued functions.

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…