Related papers: On the ternary Estermann problem with almost propo…

We address asymptotic formulae for the classical Poincar\'e-Perron problem of linear differential equations with almost constant coefficients in a half line $[t_0,+\infty)$ for high order equation $n\ge 5$ and some $t_0\in\mathbb{R}$. By…

The Painlev\'e-III equation with parameters $\Theta_0=n+m$ and $\Theta_\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\infty;m)=1$ whenever $n\in\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in…

Denote by $N(n)$ the number of integer solutions $(x_1,\,x_2,\ldots ,x_n)$ of the equation $x_1+x_2+\ldots+x_n=x_1x_2\cdot\ldots\cdot x_n$ such that $x_1\ge x_2\ge\ldots\ge x_n\ge 1$, $n \in \mathbb{Z}^+$. The aim of this paper are is…

For any $\varepsilon >0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert \alpha n + \beta \rVert < x^{-\frac{1}{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real number $x$. In…

A conjecture predicting an injective and surjective mapping $X = \displaystyle\frac{\sigma(p^k)}{p^k}, Y = \displaystyle\frac{\sigma(m^2)}{m^2}$ between OPNs $N = {p^k}{m^2}$ (with Euler factor $p^k$) and rational points on the hyperbolic…

I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular…

We improve a previous unconditional result about the asymptotic behavior of $\sum_{n\le x} r(n)r(n+m)$ with $r(n)$ the number of representations of $n$ as a sum of two squares when $m$ may vary with $x$.

Let $p$ be a prime and $n$ a positive integer such that $\sqrt{\frac p2} + 1 \leq n \leq \sqrt{p}$. For any arithmetic progression $A$ of length $n$ in $\mathbb{F}_p$, we establish an asymptotic formula for the number of directions…

The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real) forms $A$…

We prove that there are infinitely many solutions of $$ |\lambda_0+\lambda_1p+\lambda_2P_r|<p^{-\tau}, $$ where $r=3,$ $\tau=\frac1{118}$, and $\lambda_0$ is an arbitrary real number and $\lambda_1,\lambda_2\in\BR$ with $\lambda_2\neq0$ and…

We prove asymptotic results for the singular series associated to the distribution of three primes. Assuming a quantitative version of Hardy and Littlewood's conjecture on prime 3-tuples, we deduce an asymptotic formula related to the joint…

Asymptotic expansion is constructed and justified for the solution to a nonuniform Neumann boundary-value problem for the Poisson equation with the right-hand side that depends both on longitudinal and transversal variables in a thin…

We study an asymptotic formula for counting integral points over an equation defined by a non-degenerated indefinite integral ternary quadratic form $f$ representing a non-zero integer $a$ such that $-a\cdot det(f)$ is square over a number…

We obtain an asymptotic expansion for $p(n)$, the number of partitions of a natural number $n$, starting from a formula that relates its generating function $f(t), t\in (0,1)$ with the characteristic functions of a family of sums of…

We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1} (an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to 0+$, where $p$…

When asymptotically analysing the summatory function of a $q$-regular sequence in the sense of Allouche and Shallit, the eigenvalues of the sum of matrices of the linear representation of the sequence determine the "shape" (in particular…

Let $\mathbb{P}$ denote the set of primes and $\mathcal{N}\subset \mathbb{N}$ be a set with arbitrary weights attached to its elements. Set $\mathfrak{p}_{\mathcal{N}}(n)$ to be the restricted partition function which counts partitions of…

Given two finite-dimensional representations $\rho$ and $\sigma$ of $\mathsf{SU}(n)$, when is there $n \in \mathbb{N}$ such that $\rho^{\otimes n}$ is isomorphic to a subrepresentation of $\sigma^{\otimes n}$? When is there a third…

For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \in \mathbb Z^3$ of $a \frac{x(x-1)}{2} + b \frac{y(y-1)}{2} + c \frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some…

We obtain an asymptotic formula with a power-saving error term for counting the integer points $(a,b,c,d)$ in an expanding box $[-X,X]^4$ that satisfy the determinant equation $x_1x_2-x_3x_4=r$ for $r \neq 0$ with two of entries to be…