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An asymptotic interation method for solving second-order homogeneous linear differential equations of the form y'' = lambda(x) y' + s(x) y is introduced, where lambda(x) \neq 0 and s(x) are C-infinity functions. Applications to Schroedinger…

Let $a,b>0$ be coprime integers. Assuming a conjecture on Hecke eigenvalues along binary cubic forms, we prove an asymptotic formula for the number of primes of the form $ax^2+by^3$ with $x \leq X^{1/2}$ and $y \leq X^{1/3}$. The proof…

Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2\beta_k}, \quad |z|=1, \quad |a_k|=1, \quad \beta_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$ for…

We prove explicit asymptotic formulae for some functions used in sieve methods and show that there exists no odd multiperfect number of abundancy four whose squared part is cubefree.

We investigate integers whose base $g$ expansion omits a fixed digit and which can be represented as a sum of two prime squares. In the first part of the paper, we apply the Hardy--Littlewood circle method to obtain asymptotic formulas for…

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other…

We establish an asymptotic formula for the number of lattice points in the sets \[ \mathbf S_{h_1, h_2, h_3}(\lambda): =\{x\in\mathbb Z_+^3:\lfloor h_1(x_1)\rfloor+\lfloor h_2(x_2)\rfloor+\lfloor h_3(x_3)\rfloor=\lambda\} \quad…

We fix a maximal order $\mathcal O$ in $\F=\R,\C$ or $\mathbb{H}$, and an $\F$-hermitian form $Q$ of signature $(n,1)$ with coefficients in $\mathcal O$. Let $k\in\N$. By applying a lattice point theorem on the $\F$-hyperbolic space, we…

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…

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 investigate the large-distance asymptotics of optimal Hardy weights on $\mathbb Z^d$, $d\geq 3$, via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar…

In the first part of this work \cite{Du}, a quantitative supplement to the Hasse principle was given for the count of the number of automorphic orbits of primitive zeros of a genus of ternary quadratic forms. This sequel contains, for…

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses…

We consider cubic forms $\phi_{a,b}(x,y,z) = ax^3 + by^3 - z^3$ with coefficients $a,b \in \mathbb{Z}$. We give an asymptotic formula for how many of these forms are locally soluble everywhere, i.e. we give an asymptotic formula for the…

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 obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to…

Let $\mathbf{f} = (f_1, \ldots, f_R)$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_j (x_1, \ldots, x_n) = 0 \ (1…

In this paper, we derive an asymptotic approximation to the eigenvalues of the linear differential equation $$ -y"(x)+q(x)y(x)=\lambda y(x), x\in (a,b) $$ with boundary conditions of general form, when $q$ is a measurable function which has…

We establish an asymptotic formula for the weighted quantum variance of dihedral Maass forms on $\Gamma_0(D) \backslash \mathbb H$ in the large eigenvalue limit, for certain fixed $D$. As predicted in the physics literature, the resulting…

An asymptotic formula for $$ \int_{T/2}^{T}Z^2(t)Z(t+U)\,dt\qquad(0< U = U(T) \le T^{1/2-\varepsilon}) $$ is derived, where $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}\quad(t\in\Bbb R), \quad \zeta(s) = \chi(s)\zeta(1-s) $$ is…