Related papers: Second numbers in arithmetic progressions

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…

Let $A, B\subseteq \mathbb{Z}$ be finite, nonempty subsets with $\min A=\min B=0$, and let $$\delta(A,B)={\begin{array}{ll} 1 & \hbox{if} A\subseteq B, 0 & \hbox{otherwise.} If $\max B\leq \max A\leq |A|+|B|-3$ and \label{one}|A+B|\leq…

In this article we give, for the fist time the solution of the general difference equation of 2-degree. We also give as application the expansion of a continued fraction into series, which was first proved, found in the past by the author.

Let $d$ be any positive and non square integer. We prove an upper bound for the first two moments of the length $T(d)$ of the period of the continued fraction expansion for $\sqrt{d}$. This allows to improve the existing results for the…

Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric)…

We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.

In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author described periods and sometimes precise form of…

For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied…

We obtain a new bound on the second moment of modified shifted convolutions of the generalized 3-fold divisor function, and show that, for applications, the modified version is sufficient.

As the conclusion of a line of investigation undertaken in two previous papers, we compute asymptotic frequencies for the values taken by numerators of differences of consecutive Farey fractions with denominators restricted to lie in…

We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.

Given a sequence $\{b_{i}\}_{i=1}^{n}$ and a ratio $\lambda \in (0,1),$ let $E=\cup_{i=1}^n(\lambda E+b_i)$ be a homogeneous self-similar set. In this paper, we study the existence and maximal length of arithmetic progressions in $E$. Our…

We answer a number of questions of Erd\H{o}s on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a…

We find upper bounds that are sharp for the number of $k$th powers inside arbitrary arithmetic progressions whose step has $O(1)$ many divisors.

We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $\mathbb{C}^2$ and use Tauberian methods to…

In this paper we deal with composite rational functions having zeros and poles forming consecutive elements of an arithmetic progression. We also correct a result published earlier related to composite rational functions having a fixed…

We investigate a result on convergence of double sequences of numbers and how it extends to measurable functions.

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^\alpha$ for…

Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question…