Related papers: Multiplicative functions in short arithmetic progr…

Let $\|x\|$ denote the distance from $x\in\mathbb{R}$ to the nearest integer. In this paper, we prove an existence and density statement for matrices $\boldsymbol{A}\in\mathbb{R}^{m\times n}$ satisfying…

Hooley conjectured that the variance V(x;q) of the distribution of primes up to x in the arithmetic progressions modulo q is asymptotically x log q, in some unspecified range of q\leq x. On average over 1\leq q \leq Q, this conjecture is…

We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for…

Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a…

We study higher uniformity properties of the M\"obius function $\mu$, the von Mangoldt function $\Lambda$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed…

Let $f$ be a real polynomial with irrational leading co-efficient. In this article, we derive distribution of $f(n)$ modulo one for all $n$ with at least three divisors and also we study distribution of $f(n)$ for all square-free $n$ with…

In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.

We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new…

Let $f$ be a Steinhaus random multiplicative function, and for $\alpha\in \mathbb{R}$, let $d_\alpha$ denote the $\alpha$-divisor function. For $\alpha \in (1,2)$ we establish that $$ \mathbb{E}\bigg\{\Big|\frac{1}{\sqrt{x}}\sum_{n\leq x}…

Assuming a uniform $q$-variant of the prime $k$-tuple conjecture, we compute moments of the number of primes in arithmetic progressions to a large modulus $q$ as the residue classes vary. Consequently, depending on the size of $\varphi(q)$,…

In this paper, we employ a version of Stepanov's method, developed by Hanson and Petridis, to prove several results on additive irreducibility of multiplicative subgroups in finite fields of prime order $p$. Specifically, we show that if a…

Given $T\subset\mathbb{R}$ and a metric space $M$, we introduce a nondecreasing sequence of pseudometrics $\{\nu_n\}$ on $M^T$ (the set of all functions from $T$ into $M$), called the \emph{joint modulus of variation}. We prove that if two…

We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original…

When solving a number of problems in prime number theory, it is sufficient that $t(x;q)$ admits an estimate close to this one. The best known estimates for $t(x;q)$ previously belonged to G.~Montgomery, R.~Vaughn, and Z.~Kh.~Rakhmonov. In…

Fix a modulus $q$. One would expect the number of primes in each invertible residue class mod $q$ to be multinomially distributed, i.e. for each $p \,\mathrm{mod}\, q$ to behave like an independent random variable uniform on…

We fix a non-zero integer $a$ and consider arithmetic progressions $a \bmod q$, with $q$ varying over a given range. We show that for certain specific values of $a$, the arithmetic progressions $a \bmod q$ contain, on average, significantly…

We investigate when the exponential sum $S_f(x,\alpha) := \sum_{n\le x}f(n)\mathrm{e}(n\alpha)$ is bounded, for a multiplicative function $f$ and $\alpha\in\mathbb{R}$. We show that under natural assumptions, $S_f(x,\alpha)$ is bounded only…

Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short intervals? We resolve a function field variant of this problem in…

Hooley proved that if $f\in \Bbb Z [X]$ is irreducible of degree $\ge 2$, then the fractions $\{ r/n\}$, $0<r<n$ with $f(r)\equiv 0\pmod n$, are uniformly distributed in $(0,1)$. In this paper we study such problems for reducible…

We say a power series $a_0+a_1q+a_2q^2+\cdots$ is \emph{multiplicative} if $n\mapsto a_n/a_1$ for positive integers $n$ is a multiplicative function. Given the Eisenstein series $E_{2k}(q)$, we consider formal multiplicative power series…