Related papers: On Multiplicative Functions with Bounded Partial S…

In this work, approximations for real two variables function $f$ which has continuous partial $(n-1)$-derivatives $(n \ge 1)$ and has the $n$--th partial derivative of bounded bivariation or absolutely continuous are established. Explicit…

For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \[ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \\ P(n) > \sqrt{x}}} f(n), \] where…

We explore two questions about pseudo-polynomials, which are functions $f:\mathbb N \to \mathbb Z$ such that $k$ divides $f(n+k) - f(n)$ for all $n,k$. First, for certain arbitrarily sparse sets $R$, we construct pseudo-polynomials $f$ with…

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{\theta \in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n \theta) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N…

An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue…

The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supremum norm of the polynomial itself. The reversed Bernstein…

In this paper, we consider sums of three generalized $m$-gonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restrictions on $m$ modulo $30$, we show that a density one set of…

If n is a positive integer, let h(n) denote the maximal value of the product of distinct primes whose sum does not exceed n. We give some properties of this function h and describe an algorithm able to compute h(n) for large values of n.

We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower…

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\…

Based on the definition of generalized partially bent functions, using the theory of linear transformation, the relationship among generalized partially bent functions over ring Z N, generalized bent functions over ring Z N and affine…

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…

The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space $%H_{p}$ to the Lebesgue…

The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on…

We characterize the limiting behavior of partial sums of multiplicative functions $f:\mathbb{F}_q[t]\to S^1$. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined…

We obtain statistical results on the possible distribution of all partial sums of a Kloosterman sum modulo a prime, by computing explicitly the support of the limiting random Fourier series of our earlier functional limit theorem for…

It is be shown that the sequence of Bernstein polynomials for a function of several variables converges to this function uniformly along with every partial derivative of any order, provided that the latter derivative is well defined and…

We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim we prove a new…

We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the…