Narrow Results

We use recent results of Rolen, Zwegers, and the first author to study the characters of irreducible (highest weight) modules for the vertex operator algebra $L_{{𝔰𝔩}_{\ell }}(-{\mathrm{\Lambda }}_0)$. We establish asymptotic behaviors of characters for the (ordinary) irreducible $L_{{𝔰𝔩}_{\ell }}(-{\mathrm{\Lambda }}_0)$-modules. As a consequence, we prove that their quantum dimensions are one, as predicted by the representation theory. We also establish a full asymptotic expansion of irreducible characters for ${𝔰𝔩}_3$. Finally, we determine a decomposition formula for the full characters in terms of unary theta and false theta functions which allows us to study their modular properties.

Asymptotic formulas are derived for the Stieltjes–Wigert polynomials S_n(z; q) in the complex plane as the degree n grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the q-Airy function A_q(z) is used as the approximant. For real x > 1/4, a limiting relation is also established between the q-Airy function A_q(x) and the ordinary Airy function Ai(x) as q → 1.

We present an asymptotic formula for the number of line segments connecting q + 1 points of an n × n square grid, and a sharper formula, assuming the Riemann hypothesis. We also present asymptotic formulas for the number of lines through at least q points and, respectively, through exactly q points of the grid. The well-known case q = 2 is so generalized.

We are going to study the mean values of some multiplicative functions connected with the divisor function in short interval of summation. The asymptotics for such mean values will be proved. Considering instead of well-known multiplicative functions, their inverses lead to very weak results of application of standard methods of complex integration. In order to get better estimations, we propose another method which uses as its main tools the density estimates and zero-free region for Riemann ζ-function and Dirichlet L-functions.

For each positive integers n, let g(n) be the number of arithmetic expressions evaluating to n and involving only the constant 1, additions and multiplications, with the restriction that multiplication by 1 is not allowed. We consider two arithmetic expressions to be equal if one can be obtained from the other through a repeated application of the commutative and associative properties. We give an algorithm to compute g(n) and prove that , as n → +∞, where β ≔ log(24)/24.

In this paper, we deduce the asymptotic formulas of parametric digamma function $\mathrm{\Psi }(-s;a)$ at the integers and poles. Then using these identities and residue theorem, we establish a large number of formulas of double series involving parametric harmonic numbers. Some illustrative special cases as well as immediate consequences of the main results are also considered.

We study the distribution of the generalized gcd and lcm functions on average. The generalized gcd function, denoted by ${(m,n)}_b$, is the greatest $b$th power divisor common to $m$ and $n$. Likewise, the generalized lcm function, denoted by ${[m,n]}_b$, is the smallest $b$th power multiple common to $m$ and $n$. We derive asymptotic formulas for the average order of the arithmetic, geometric, and harmonic means of ${(m,n)}_b$. Additionally, we also deduce asymptotic formulas with error terms for the means of ${(n_1,n_2,\dots ,n_k)}_b$, and ${[n_1,n_2,\dots ,n_k]}_b$ over a set of lattice points, thereby generalizing some of the previous work on gcd and lcm-sum estimates.

Asymptotic formulas are derived for the Stieltjes–Wigert polynomials S_n(z;q) in the complex plane as the degree n grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the q-Airy function A_q(z) is used as the approximant. For real x > 1/4, a limiting relation is also established between the q-Airy function A_q(x) and the ordinary Airy function Ai(x) as q → 1.