An Asymptotic Formula for the Number of Smooth Values of a Polynomial

The asymptotic formulae of sums of two smooth squares for divisor function

A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. Let $s_{1}$ and $s_{2}$ are $y$-smooth numbers. We consider sums of smooth squares of the binary Titchmarsh divisor problem and give asymptotic formulae for $\sum_{s_{1}^{2}+s_{2}^{2}\le x}τ(s_{1}^{2}+s_{2}^{2}+1)$ for $(\log x)^{K}\le y<x^{\frac{1}{2}}$, where $K$ is large enough.

The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers

In this paper we present and prove rapidly convergent formulas for the distribution of the $3$-smooth, $5$-smooth, $7$-smooth and all other smooth numbers. One of these formulas is another version of a formula due to Hardy and Littlewood for the arithmetic function $N_{a,b}(x)$, which counts the number of positive integers of the form $a^{p}b^{q}$ less than or equal to $x$.

短区间内光滑数的渐近性

Asymptotics for smooth numbers in short intervals

本文研究短区间中 $y$-光滑数的密度渐近，利用 Perron 积分、围道平移与 zeta 零点密度估计，无条件及在 Riemann 假设下分别扩大了结果适用的 θ 与 y 范围。

Smooth arithmetical sums over k-free integers

We use partial zeta functions to analyse the asymptotic behaviour of certain smooth arithmetical sums over smooth k-free integers.

Smooth numbers in Beatty sequences

An asymptotic formula is given for the number of y-smooth numbers up to x in a Beatty sequence corresponding to an irrational number of finite type.

A New Effective Asymptotic Formula for the Stieltjes Constants

We derive a new integral formula for the Stieltjes constants. The new formula permits easy computations as well as an exact approximate asymptotic formula. Both the sign oscillations and the leading order of growth are provided. The formula can also be easily extended to generalized Euler constants.