Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Let $m,n\ge 2$, $m\le n$. It is well-known that the number of (two-dimensional) threshold functions on an $m\times n$ rectangular grid is {eqnarray*} t(m,n)=\frac{6}{\pi^2}(mn)^2+O(m^2n\log{n})+O(mn^2\log{\log{n}})= \frac{6}{\pi^2}(mn)^2+O(mn^2\log{m}). {eqnarray*} We improve the error term by showing that $$ t(m,n)=\frac{6}{\pi^2}(mn)^2+O(mn^2). $

The arithmetic derivative and Leibniz-additive functions

An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$f(mn)=f(m)h_f(n)+f(n)h_f(m)$$ for all positive integers $m$ and $n$. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative $D$; namely, $D$ is Leibniz-additive with $h_D(n)=n$. In this paper, we study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function $f$ is totally determined by the values of $f$ and $h_f$ at primes. We also consider properties of Leibniz-additive functions with respect to the usual product, composition and Dirichlet convolution of arithmetic functions

Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative

Let $pinmathbb P$ and $sinmathbb R$, and suppose that$emptysetne Psubsetmathbb P$ is finite.Given $ninmathbb Z_+$, let $n\u27$, $n\u27_p$, and $n\u27_P$ denote respectively its arithmetic derivative, arithmetic partial derivative with respect to~$p$,and arithmetic subderivative with respect to~$P$. We study the asymptotics of $$sum_{1le nle x}frac{n\u27}{n^s},,sum_{1le nle x}frac{n\u27_p}{n^s},quad{rm and},,sum_{1le nle x}frac{n\u27_P}{n^s}.$$ We also show that the abscissa of convergence of the corresponding Dirichlet series equals~two

Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative

Let $pinmathbb P$ and $sinmathbb R$, and suppose that$emptysetne Psubsetmathbb P$ is finite.Given $ninmathbb Z_+$, let $n\u27$, $n\u27_p$, and $n\u27_P$ denote respectively its arithmetic derivative, arithmetic partial derivative with respect to~$p$,and arithmetic subderivative with respect to~$P$. We study the asymptotics of $$sum_{1le nle x}frac{n\u27}{n^s},,sum_{1le nle x}frac{n\u27_p}{n^s},quad{rm and},,sum_{1le nle x}frac{n\u27_P}{n^s}.$$ We also show that the abscissa of convergence of the corresponding Dirichlet series equals~two

Asymptotics for numbers of line segments and lines in a square grid

We present an asymptotic formula for the number of line segments connecting q+1 points of an nxn 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

A Survey on the Permanence of Finnish Students' Arithmetical Skills and the Role of Motivation

This study concerns the permanence of the basic arithmetical skills of Finnish students by investigating how a group ( = 463) of the eighth and eleventh year students and the university students of humanities perform in problems that are slightly modified versions of certain PISA 2003 mathematics test items. The investigation also aimed at finding out what the impact of motivation-related constructs, for example, students' achievement goal orientations, is and what their perceived competence beliefs and task value on their performance in mathematics are. According to our findings, the younger students' arithmetical skills have declined through the course of ten years but the older students' skills have become generic to a greater extent. Further, three motivational clusters could be identified accounting for 7.5 per cent of students' performance in the given assignments. These results are compatible with the outcomes of the recent assessments of the Finnish students' mathematical skills and support the previous research on the benefits of learning orientation combined with the high expectation of success and the valuing of mathematics learning

Lower bounds for the largest eigenvalue of the gcd matrix on {1,2,…,n}\{1,2,\dots ,n\}

summary:Consider the $n\times n$ matrix with $(i,j)$'th entry $\gcd {(i,j)}$. Its largest eigenvalue $\lambda _n$ and sum of entries $s_n$ satisfy $\lambda _n>s_n/n$. Because $s_n$ cannot be expressed algebraically as a function of $n$, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that $\lambda _n>6\pi ^{-2}n\log {n}$ for all $n$. If $n$ is large enough, this follows from F. Balatoni (1969)