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The goal of this article is to study the discrepancy of the distribution of arithmetic sequences in arithmetic progressions. We will fix a sequence $\A=\{\a(n)\}_{n\geq 1}$ of non-negative real numbers in a certain class of arithmetic…

We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression.…

Let $f$ be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish an asymptotical formula for the quantity \[ S_f(x):=\sum_{n\leq x}\frac{f([x/n])}{[x/n]} \] as $x\rightarrow\infty$, where $[t]$ is…

We study the distribution of partition parts in arithmetic progressions and find asymptotic results that capture all exponentially growing terms. This is accomplished by studying the behavior of non-modular Eisenstein series that appear in…

We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S\subseteq\mathbb{Z}_q^n$ containing $|S|=\mu\cdot q^n$ elements must contain at…

Let $x$ be a positive integer. We give an asymptotic formula for the number of primes in the set $\{\fl{x/n}, 1 \le n \le x\}$ and give some related results.

Given a countably infinite group $G$ acting on some space $X$, an increasing family of finite subsets $G_n$ and $x\in X$, a natural question to ask is what asymptotical distribution the sets $G_nx$ form. More formally, we define for a…

By a classical theorem of Koksma the sequence of fractional parts $(\{x^n\})_{n \geq 1}$ is uniformly distributed for almost all values of $x$. In the present paper we obtain an exact quantitative version of Koksma's theorem, by calculating…

By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided…

Motivated by recent results, we study sums of the form $S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor \right)$, where $f$ is an arithmetic function and $\left\lfloor\cdot\right\rfloor$ denotes the greatest integer…

We study the sums $$ S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor\right) $$ when $f$ is supported on $r$th powers with $r\geq 2$. This restriction allows us to give nontrivial estimates for one of the error terms in…

Let [t] be the integral part of the real number t and let 1 P be the characteristic function of the primes. Denote by $\pi$ G (x) the number of primes in the floor function set G(x) := {[ x n ] : 1 n x} and by S 1 P (x) the number of primes…

Let $S_m f$ denote the $m$-th partial sum of the Walsh-Fourier series of $f \in L^1$. For an increasing sequence $a=(a(n))_{n \geq 1}$ of positive integers, consider the arithmetic means $$ \sigma_N f:=\frac{1}{N} \sum_{n=1}^N S_{a(n)} f .…

Let $\left[x\right]$ be the largest integer not exceeding $x$. For $0<\theta \leq 1$, let $\pi_{\theta}(x)$ denote the number of integers $n$ with $1 \leq n \leq x^{\theta}$ such that $\left[\frac{x}{n}\right]$ is prime and…

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special…

To allow for Division By Zero, we develop a new algebraic structure containing addition and multiplication called an S-Extension of a Field. This unique structure extends a Field so that the equation $0\cdot s=x$ has exactly one solution…

For any $\varepsilon >0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert \alpha n + \beta \rVert < x^{-\frac{1}{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real number $x$. In…

Let $\mathcal{T}$ be a collection of 3-element subsets $S$ of $\{1, \ldots,n\}$ with the property that if $i<j<k$ and $a<b<c$ are two 3-element subsets in $S$, then there exists an integer sequence $x_1 < x_2 < \cdots < x_n$ such that $x_i,…

We prove a central limit theorem for the joint distribution of $s_q(A_jn)$, $1\le j \le d$, where $s_q$ denotes the sum-of-digits function in base~$q$ and the $A_j$'s are positive integers relatively prime to $q$. We do this in fact within…

Let a be an integer and q a prime number. In this paper, we find an asymptotic formula for the number of positive integers n < x with the property that no divisor d > 1 of n lies in the arithmetic progression a modulo q.