Related papers: On a certain additive divisor problem

We obtain asymptotics with a power saving error term for $\int_0^1|\sum_{n\le X} d(n)e(n\alpha)|^sd\alpha$ for $s < 2$, where $d(n) = \sum_{d | n} 1$ is the divisor function. We also obtain such asymptotics for $\int_0^1|\sum_{n\le…

We study a mean value of the classical additive divisor problem. The main term we are interested in here is the one by Motohashi, but we also give an upper bound for the case where the main term is that of Atkinson. Furthermore, we point…

We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the…

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focussing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the…

This paper is concerned with asymptotic theory for penalized spline estimator in bivariate additive model. The focus of this paper is put upon the penalized spline estimator obtained by the backfitting algorithm. The convergence of the…

In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.

We propose a multivariate probability distribution that models a linear correlation between binary and continuous variables. The proposed distribution is a natural extension of the previously developed multivariate binary distribution. As…

A semilinear reaction-diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter $\eps^2$, is considered. It can have multiple solutions. An asymptotic expansion is constructed for…

A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. Let $s_{1}$ and $s_{2}$ are $y$-smooth numbers. We consider sums of smooth squares of the binary Titchmarsh divisor problem and give asymptotic…

Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a $d^{p+q}$-dimensional matrix variable that…

In this paper we propose an iterative method using alternating direction method of multipliers (ADMM) strategy to solve linear inverse problems in Hilbert spaces with general convex penalty term. When the data is given exactly, we give a…

We obtain an asymptotic formula for the average value of the divisor function over the integers $n \le x$ in an arithmetic progression $n \equiv a \pmod q$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In…

Let $\Delta(x)$ be the error term of the Dirichlet divisor problem. The asymptotic formula of the integral $\int_1^T\Delta^k(x)dx$ is established for any integer $3\leq k\leq 9$ by an unified method. Similar results are also established for…

We prove that a linear fractional operator with an asymptotically constant lower order term in the whole space admits eigenvalues.

This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper…

We introduce a new technique for solving uni-parametric versions of linear programs, convex quadratic programs, and linear complementarity problems in which a single parameter is permitted to be present in any of the input data. We…

We prove some distribution results for the $k$-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length $X$ of the sum, with appropriate constrains and averaging on the moduli, saving a power of $X$…

Let X be a smooth complex projective variety of dimension d. It is classical that ample line bundles on X satisfy many beautiful geometric, cohomological, and numerical properties that render their behavior particularly tractable. By…

We extend a recently established asymptotic normality theorem for generalized linear mixed models to include the dispersion parameter. The new results show that the maximum likelihood estimators of all model parameters have asymptotically…

We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we had introduced the distinction between periodic and…