Related papers: Asymptotics for some polynomial patterns in the pr…

The Fourier coefficients of a Maass form $\phi$ for SL$(n,\mathbb Z)$ are complex numbers $A_\phi(M)$, where $M=(m_1,m_2,\ldots,m_{n-1})$ and $m_1,m_2,\ldots ,m_{n-1}$ are nonzero integers. It is well known that coefficients of the form…

In 2002, M.Ram Murty showed that if p is a prime with k-adic expansion :$p = \sum_{i = 0}^n a_i k^i$ , then the polynomial $f(x) = \sum_{i = 0}^n a_ix^i$ is irreducible in $\mathbb{Z}[x]$.When $k = 10$ , it's a result of A.Cohn. I think…

We present an informal review of results on asymptotics of orthogonal polynomials, stressing their spectral aspects and similarity in two cases considered. They are polynomials orthonormal on a finite union of disjoint intervals with…

We introduce the sequence $(a_n) \subset (0,1]$ and prove that the asymptotic behaviour of $\sum_{k=1}^n a_k$ is the same than $\pi(n)$, the prime-counting function. We also obtain that $\pi(n) \sim n a_n$ and we estimate…

We consider the semi-classical generalized Freud weight function \[w_{\lambda}(x;t) = |x|^{2\lambda+1}\exp(-x^4 +tx^2),\qquad x\in\mathbb{R},\] with $ \lambda>-1$ and $t\in\mathbb{R}$ parameters. We analyze the asymptotic behavior of the…

In this paper we study the asymptotic behaviour of weighted random sums when the sum process converges stably in law to a Brownian motion and the weight process has continuous trajectories, more regular than that of a Brownian motion. We…

Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on GL(1)) was used by Dirichlet to prove…

Our goal is to find an asymptotic behavior as $n\to\infty$ of the orthogonal polynomials $P_{n}(z)$ defined by Jacobi recurrence coefficients $a_{n}$ (off-diagonal terms) and $ b_{n}$ (diagonal terms). We consider the case $a_{n}\to\infty$,…

Let $ a_1(x)p_1(x)^n + \cdots + a_k(x)p_k(x)^n $ as well as $ b_1(x)q_1(x)^m + \cdots + b_l(x) q_l(x)^m $ be two polynomial power sums where the complex polynomials $ p_i(x) $ and $ q_j(x) $ are all non-constant. Then in the present paper…

Let $\mathfrak{p}_{\mathbb{P}_r}(n)$ denote the number of partitions of $n$ into $r$-full primes. We use the Hardy-Littlewood circle method to find the asymptotic of $\mathfrak{p}_{\mathbb{P}_r}(n)$ as $n \to \infty$. This extends previous…

We prove several results regarding the distribution of numbers that are the product of a prime and a $k$-th power. First, we prove an asymptotic formula for the counting function of such numbers; this generalises a result of E. Cohen. We…

Our goal is to find an asymptotic behavior as $n\to\infty$ of orthogonal polynomials $P_{n}(z)$ defined by the Jacobi recurrence coefficients $a_{n}, b_{n}$. We suppose that the off-diagonal coefficients $a_{n}$ grow so rapidly that the…

Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the…

Let $P^+(n)$ denote the largest prime of the integer $n$. Using the \begin{align*}\Psi\_{F\_1\cdots F\_t}\left(\mathcal{K}\cap[-N,N]^d,N^{1/u}\right):=\\#\left\{\mathcal{K}\in…

In this work we prove a prime number type theorem involving the normalised Fourier coefficients of holomorphic and Maass cusp forms, using the classical circle method. A key point is in a recent paper of Fouvry and Ganguly, based on…

Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{\nu n}$ with $\nu>1/2$.

We apply the Euler--Maclaurin formula to find the asymptotic expansion of the sums $\sum_{k=1}^n (\log k)^p / k^q$, ~$\sum k^q (\log k)^p$, ~$\sum (\log k)^p /(n-k)^q$, ~$\sum 1/k^q (\log k)^p $ in closed form to arbitrary order ($p,q…

Let $\mathbb{F}_q[t]$ be the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements, and let $p$ be the characteristic of $\mathbb{F}_q$. We denote $\widetilde{G}_q(k)$ to be the least integer $t_0$ with the property that…

Recently, Donoso, Le, Moreira and Sun studied the asymptotic behavior of the averages of completely multiplicative functions over the Gaussian integers. They derived Wirsing's theorem for Gaussian integers, answered a question of…

We present a formula describing the asymptotics of a class of multivariate orthogonal polynomials with hyperoctahedral symmetry as the degree tends to infinity. The polynomials under consideration are characterized by a factorized weight…