Related papers: On further questions regarding unit fractions

We prove that the ratio of the Newman sum over numbers multiple of a fixed integer which is not multiple of 3 and the Newman sum over numbers multiple of a fixed integer divisible by 3 is o(1) when the upper limit of summing tends to…

Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2.…

We consider two problems regarding some divisibility properties of the subset sums of a set $A\subseteq \{1, 2, \ldots ,n\}$. At the beginning, we study the cardinality of $A$ which has the following property: For every $d\le n$ there is a…

For nonempty sets $A,B$ of nonnegative integers and an integer $n$, let $r_{A,B}(n)$ be the number of representations of $n$ as $a+b$ and $d_{A,B}(n)$ be the number of representations of $n$ as $a-b$, where $a\in A, b\in B$. In this paper,…

For $d\geq 2$ and any norm on $\mathbb R^d$, we prove that there exists a set of $n$ points that spans at least $(\tfrac d2-o(1))n\log_2n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon,…

Consider the positive integers $n$ such that $n$ divides the $n$-th Fibonacci number, and their counting function $A$. We prove that \[A(x) \leq x^{1-(1/2+o(1))\log\log\log x/\log\log x}.\]

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

Euler gives a continued fraction representation of (1 + x)n. involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for "vanishing" n, and for infinite…

We show that every set $A$ of natural numbers with positive upper density can be shifted to contain the restricted sumset $\{b_1 + b_2 : b_1, b_2\in B \text{ and } b_1 \neq b_2 \}$ for some infinite set $B \subset A$.

We consider a group-theoretic analogue of the classic subset sum problem. In this brief note, we show that the subset sum problem is NP-complete in the first Grigorchuk group. More generally, we show NP-hardness of that problem in weakly…

A strictly increasing sequence $\mathscr{A}$ of positive integers is said to be primitive if no term of $\mathscr{A}$ divides any other. Erd\H{o}s showed that the series $\sum_{a \in \mathscr{A}} \frac{1}{a \log a}$, where $\mathscr{A}$ is…

Given a sequence $\mathscr{A}=\{a_0<a_1<a_2\ldots\}\subseteq \mathbb{N}$, let $r_{\mathscr{A},h}(n)$ denote the number of ways $n$ can be written as the sum of $h$ elements of $\mathscr{A}$. Fixing $h\geq 2$, we show that if $f$ is a…

Building on techniques recently introduced by the second author, and further developed by the first author, we show that a positive integer $N$ may be rigorously and deterministically factored into primes in at most \[ O\left( \frac{N^{1/5}…

In this paper we show that if $A$ is a subset of the primes with positive relative density $\delta$, then $A+A$ must have positive upper density $C_1\delta e^{-C_2(\log(1/\delta))^{2/3}(\log\log(1/\delta))^{1/3}}$ in $\mathbb{N}$. Our…

A new, constructive proof with a small explicit constant is given to the Erd\H{o}s-Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at…

Let $G\cong \mathbb Z/m_1\mathbb Z\times\ldots\times \mathbb Z/m_r\mathbb Z$ be a finite abelian group with $m_1\mid\ldots\mid m_r=\exp(G)$. The $n$-term subsums version of Kneser's Theorem, obtained either via the DeVos-Goddyn-Mohar…

Let $Z(t)=\chi^{-1/2}(1/2+it)\zeta(1/2+it)=e^{i\theta(t)}\zeta(1/2+it)$ be Hardy's function and $g(n)$ be the $n$-th Gram points defined by $\theta(g(n))=\pi n$. Titchmarsh proved that $\sum_{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log^{3/4}N) $…

Let $d(n)$ denote the number of divisors of a positive integer $n$. A classical problem in analytic number theory is given by the asymptotic behavior of the divisor sum $\sum_{n \leq x} \frac{1}{d(n)}$, with Ramanujan having introduced an…

For a pair of distinct non-CM newforms of weights at least 2, having rational integral Fourier coefficients $a_{1}(n)$ and $a_{2}(n)$, under GRH, we obtain an estimate for the set of primes $p$ such that $$ \omega(a_1(p)-a_2(p)) \le […

In this paper we prove new upper bounds for the sum $\sum_{n=a+1}^{a+N}f(n)$, for a certain class of arithmetic functions $f$. Our results improve the previous results of G. Bachman and L. Rachakonda.