Related papers: On Multiplicative Functions with Bounded Partial S…

We obtain almost sure bounds for the weighted sum $\sum_{n \leq t} \frac{f(n)}{\sqrt{n}}$, where $f(n)$ is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated…

Tur\'an observed that logarithmic partial sums $\sum_{n\le x}\frac{f(n)}{n}$ of completely multiplicative functions (in the particular case of the Liouville function $f(n)=\lambda(n)$) tend to be positive. We develop a general approach to…

We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite fields analogues…

Let $G_n= \prod_{k=0}^n \binom{n}{k},$ the product of the elements of the $n$-th row of Pascal's triangle. This paper studies the partial factorizations of $G_n$ given by the product $G(n,x)$ of all prime factors $p$ of $G_n$ having $p \le…

For a large prime $p$, a rational function $\psi \in F_p(X)$ over the finite field $F_p$ of $p$ elements, and integers $u$ and $H\ge 1$, we obtain a lower bound on the number consecutive values $\psi(x)$, $x = u+1, \ldots, u+H$ that belong…

The problem is considered as to whether a monotone function defined on a subset P of a Euclidean space can be strictly monotonically extended to the whole space. It is proved that this is the case if and only if the function is {\em…

Given polynomials $f_1,\ldots,f_n$ in $m$ variables with integral coefficients, we give upper bounds for the number of integral $m$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_n$ of bounded height such that $f_1(\mathbf{u}_1), \ldots,…

We study the possibility of splitting any bounded analytic function with singularities in a closed set E union F as a sum of two bounded analytic functions with singularities in E and F respectively. We obtain some results under geometric…

We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In…

We give extensions of results on nonnegative matrix semigroups which deduce finiteness or boundedness of such semigroups from the corresponding local properties, e.g., from finiteness or boundedness of values of certain linear functionals…

For a transcendental entire function, a partial affirmative answer to Baker's question on the boundedness of its Fatou components is given. In addition, we have addressed Wang's question on Fej\'er gaps. Certain results about functions with…

We show that there is significant cancellation in certain exponential sums over small multiplicative subgroups of finite fields, giving an exposition of the arguments by Bourgain and Chang.

In this paper we derive characterizations of boundedness of the subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces when $ 0<p<1 $. Moreover, we find necessary and sufficient conditions for the…

A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this paper, we prove in $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice)…

Assuming that $S$ is the space of functions of regular variation, $\omega\in S$, $0< p<\infty$, a function $f$ holomorphic in $B^n$ is said to be of Besov space $B_p(\omega)$ if $$\|f\|^p_{B_p(\omega )}=\int_{B^n}…

We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…

The paper considers the properties of pseudo stationarity in a broad sense and pseudo strong mixing for sequences of random variables corresponding to arithmetic functions. Assertions on this topic have been proven. The implementation of…

Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…

Let $f$ be a positive multiplicative function and let $k\geq 2$ be an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in the sense that $\sum_{p}|f(p)-1|=\infty$, there exists…

Completely multiplicative functions whose sum is zero ($CMO$).The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits\_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all…