Related papers: Correlation of multiplicative functions over funct…
We use a function field analogue of a method of Selberg to derive an asymptotic formula for the number of (square-free) monic polynomials in $\mathbb{F}_q[X]$ of degree $n$ with precisely $k$ irreducible factors, in the limit as $n$ tends…
For the quantum integer $[n]_q = 1+q+...+q^{n-1}$ there is a natural polynomial multiplication $*_q$ such that $[m]_q *_q [n]_q = [mn]_q$. This multiplication leads to the functional equation $f_{mn}(q) = f_m(q)f_n(q^m),$ defined on a given…
Let $\xi_1,\xi_2,...$ be independent identically distributed random variables and $F:\bbR^\ell\to SL_d(\bbR)$ be a Borel measurable matrix-valued function. Set $X_n=F(\xi_{q_1(n)},\xi_{q_2(n)},...,\xi_{q_\ell(n)})$ where $0\leq…
We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$…
The paper considers asymptotics of summation functions of additive and multiplicative arithmetic functions. We also study asymptotics of summation functions of natural and prime arguments. Several assertions on this subject are proved and…
This paper analyzes over 30 types of q-series and the asymptotic behavior of their expansions. A method is described for deriving further asymptotic formulas using convolutions of generating functions with subexponential growth. All…
In the present note we prove an asymptotically tight relation between additive and multiplicative complexity of Boolean functions with respect to implementation by circuits over the basis {+,*,1}.
We investigate a Riemann-Hilbert problem (RHP), whose solution corresponds to a group of $q$-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the…
The paper considers estimates for the asymptotics of summation functions of bounded multiplicative arithmetic functions. Several assertions on this subject are proved and examples are considered.
Asymptotic Pade-approximant methods are utilized to estimate the leading-order unknown (i.e., not-yet-calculated) contributions to the perturbative expansions of two-current QCD correlation functions obtained from scalar-channel fermion and…
The number A(q) shows the asymptotic behaviour of the quotient of the number of rational points over the genus of non-singular absolutely irreducible curves over a finite field Fq. Research on bounds for A(q) is closely connected with the…
This article derives full asymptotic expansions for integrals of the form \[ \int_{0}^{1}f(u)(1+q\cdot u^{n})^{w/n}du \] as $n\rightarrow\infty$, with parameters real $w\neq 0$ and $q\in(-1,1]$, or positive $w$ for $q=-1$. We relate the…
Let $F\in\mathbb{Z}[x,y]$ be some polynomial of degree 2. In this paper we find the asymptotic behaviour of the least common multiple of the values of $F$ up to $N$. More precisely, we consider $\psi_F(N) =…
In earlier work, we introduced three families of polynomials where the generating function of each set includes one of the three Jackson $q$-analogs of the Bessel function. This paper gives determinant representation for each family, their…
We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial…
We consider a sequence $\{f(p)\}_{p\ {\rm prime}}$ of independent random variables taking values $\pm 1$ with probability $1/2$, and extend $f$ to a multiplicative arithmetic function defined on the squarefree integers. We investigate upper…
We give the asymptotic behavior of the ratio of two neighboring multiple orthogonal polynomials under the condition that the recurrence coefficients in the nearest neighbor recurrence relations converge.
Two-dimensional Coulomb gases on an annulus at a special inverse temperature $\beta = 2$ are studied by using the orthogonal polynomial method borrowed from the theory of random matrices. The correlation functions among the Coulomb gas…
An asymptotic formula for the sum $\sum L(1,\chi)$ is established for a family of hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_q$ as $g\rightarrow\infty$ making use of the analogue of the approximate functional…
We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$.…