Related papers: Upper bound for the generalized repetition thresho…

A nonempty finite set of positive integers A is relatively prime if gcd(A) = 1 and it is relatively prime to n if gcd(A [ fng) = 1. The number of nonempty subsets of A which are relatively prime to n is \Phi(A, n) and the number of such…

We show that for each positive integer $a$ there exist only finitely many prime numbers $p$ such that $a$ appears an odd number of times in the period of continued fraction of $\sqrt{p}$ or $\sqrt{2p}$. We also prove that if $p$ is a prime…

Sharpening (a particular case of) a result of Szemeredi and Vu and extending earlier results of Sarkozy and ourselves, we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset…

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…

In this paper, we exhibit upper and lower bounds with explicit constants for some objects related to entire $L$-functions in the critical strip, under the generalized Riemann hypothesis. The examples include the entire Dirichlet…

A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer and a square is a word of the form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares…

An \itbf{indeterminate string} (or, more simply, just a \itbf{string}) $\s{x} = \s{x}[1..n]$ on an alphabet $\Sigma$ is a sequence of nonempty subsets of $\Sigma$. We say that $\s{x}[i_1]$ and $\s{x}[i_2]$ \itbf{match} (written $\s{x}[i_1]…

We establish a law of the iterated logarithm (LIL) for the set of real numbers whose $n$-th partial quotient is bigger than $\alpha_n$, where $(\alpha_n)$ is a sequence such that $\sum 1/\alpha_n$ is finite. This set is shown to have…

Let $A$ be a set in an abelian group $G$. For integers $h,r \geq 1$ the generalized $h$-fold sumset, denoted by $h^{(r)}A$, is the set of sums of $h$ elements of $A$, where each element appears in the sum at most $r$ times. If…

In this note, we are interested in obtaining uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$. We obtain unconditional upper bounds $O(\frac{y}{\log y})$ and $O(y^{11/12})$ for all powerful numbers and…

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…

We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function $f(r)$ such that, for any set $C$ of $r$-state characters, $C$ is compatible if and only…

Let $f$ be a real-valued $1$-bounded multiplicative function. Suppose that the mean-value of $f^{2}$ exists, and $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n)e^{2\pi i n \alpha} \Big | d \alpha\leq N^{o(1)}$$ as $N \rightarrow \infty$, then…

The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…

Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}_r=\big\{1^r, 2^r, 3^r,\cdots\big\}$ the set of $r$-th powers, where $r\ge 2$ is a natural number. Let $\mathcal{W}_r$ be an additive complement of $\mathcal{S}_r$ and $$…

A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special. We investigate the structure of…

Let $(\mathcal{O}_n, \mathfrak{m})$ denote the ring of germs of holomorphic functions $\mathbb{C}^n\to \mathbb{C}$, and let $I\subseteq \mathcal{O}_n$ be an $\mathfrak{m}$-primary ideal. Demailly and Pham showed that $\mathrm{lct}(I) \geq…

For a given $x$ we consider the minimum of $\sum_{n\le x} \chi(n)/n$ as $\chi$ ranges over all quadratic Dirichlet characters. For all large $x$, this minimum is negative and we give upper and lower bounds for it.

A border of a string is a non-empty proper prefix of the string that is also a suffix. A string is unbordered if it has no border. The longest unbordered factor is a fundamental notion in stringology, closely related to string periodicity.…

We define a quantity $c_m(n,k)$ as a generalization of the notion of the composition of the positive integer $n$ into $k$ parts. We proceed to derive some known properties of this quantity. In particular, we relate two partial Bell…