Related papers: Small values of signed harmonic sums

For any graph $G=(V,E)$ with maximum degree $\Delta$ and without isolated edges, and a positive integer $r$, by $\chi'_{\Sigma,r}(G)$ we denote the $r$-distant sum distinguishing index of $G$. This is the least integer $k$ for which a…

In 2017 Tao proposed a variant Sarnak's M\"{o}bius disjointness conjecture with logarithmic averaging: For any zero entropy dynamical system $(X,T)$, $\frac{1}{\log N} \sum_{n=1} ^N \frac{f(T^n x) \mu (n)}{n}= o(1)$ for every $f\in…

For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property…

Let $\tau_k$ be the $k$-fold divisor function. By constructing an approximant of $\tau_k$, denoted as $\tau_k^*$, which is a normalized truncation of the $k$-fold divisor function, we prove that when $\exp\left(C\log^{1/2}X(\log\log…

Given $A\subseteq\mathbb Z_n$, the constant $C_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence having consecutive terms. The value of…

We show that for every $n \in \mathbb N$ there is a collection of points $p_1, \ldots, p_n$ and lines $\ell_1, \ldots, \ell_n$ in the unit square such that for any $i$ we have $p_i \in \ell_i$ and the distance from $p_i$ to any other line…

An integer $n\ge 1$ is said to be practical if every natural number $ m \le n$ can be expressed as a sum of distinct positive divisors of $n$. The number of practical numbers up to $x$ is asymptotic to $c x/\log x$, where $c$ is a constant.…

For all odd positive integers $m$, we construct $\mu$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^3,$ with $\mu\in(m,m+1)$. For $m$ large, $\mu-m$ converges to $1$, so $\mu\neq m+\tfrac 1 2$. The restriction to odd…

Let $l\geq 6$ be any integer, where $l\equiv 2$ mod $4$. Suppose that $\mu(\tau)d\tau$ is a measure with bounded variation and is supported on a compact subset of the complex plane, where…

Let $f(n)$ denote the number of unordered factorizations of a positive integer $n$ into factors larger than $1$. We show that the number of distinct values of $f(n)$, less than or equal to $x$, is at most $\exp \left( C \sqrt{\frac{\log…

A positive integer $N$ is \emph{palindromic in the base $b$} when $N = \sum_{i=0}^{k} c_i b^i$, $c_k\neq 0$,and $c_i=c_{k-i},\; i=0,1,2,...,k$, Focusing on powers of 2, we investigate the smallest base $b$ when $N=2^n$ is palindromic in the…

Inspired by a question asked on the list {\tt mathfun}, we revisit {\em Kempner-like series}, i.e., harmonic sums $\sum' 1/n$ where the integers $n$ in the summation have ``restricted'' digits. First we give a short proof that $\lim_{k \to…

Let $x\geq 1$ be a large integer, and let $\mu:\mathbb{N}\longrightarrow\{-1,0,1\}$ be the Mobius function. This article proposes an effective asymptotic result for the autocorrelation function $\sum_{n \leq x} \mu(n) \mu(n+t) =O\left(…

For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset…

We obtain an asymptotic formula for the L^1 norm of the exponential sum $M(\alpha) = \sum_{n\le X}\tau(n)e(n\alpha)$ where $\tau(n) = \sum_{d|n} 1$ is the divisor function. In particular, we show that it is $\sim C\sqrt{X}\log X$ with $C =…

Let $n\geq\nu$, let $T$ be an $n$-vertex tree with bipartition class sizes $t_1\geq t_2$, and let $S$ be a $\nu$-vertex tree with bipartition class sizes $\tau_1\geq\tau_2$. Using four natural constructions, we show that the Ramsey number…

Let $N$ be a positive number. We give an asymptotic formula for the sum of $\tau(\gcd(a,b))$ for all $a$ and $b$ with $ab \le N$.

Let $f(n)=\min_{p} |n-p|$, where $p$ is a prime. We show that there is a positive constant $\delta$ such that for any large integer $N$ there exist two positive integers $n_1$ and $n_2$ such that $N=n_1 + n_2$ and $f(n_i)\gg \ln N (\ln\ln…

We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…

A natural number $n$ is said to be $k$- multiperfect number if $\sigma (n) = k\cdot n$ for some integer $k>2$. In this paper, I will provide a lower bound for $\tau(n)$ of any $k$- multiperfect numbers. The lower bound for $\tau(n)$ will…