Related papers: On Incidences of $\varphi$ and $\sigma$ in the Fun…

We prove simple theorems concerning the maximal order of a large class of multiplicative functions. As an application, we determine the maximal orders of certain functions of the type $\sigma_A(n)= \sum_{d\in A(n)} d$, where A(n) is a…

We study a certain class of arithmetic functions that appeared in Klurman's classification of $\pm 1$ multiplicative functions with bounded partial sums, c.f., Comp. Math. 153 (8), 2017, pp. 1622-1657. These functions are periodic and…

Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \varphi(n)$ be the Euler totient function. The asymptotic formula for the new finite sum over the primes $ \sum_{p\leq…

The purpose of this article is two-folds. Firstly, we establish two sufficient conditions under which the sequence $\{f(n)\pmod{m}: n\geq1\}$ is non-periodic, where $f(n)$ is an arithmetic function. As consequences, we deduce that the…

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…

Consider the operator $E$ on arithmetic functions such that $Ef$ is the multiplicative arithmetic function defined by $(Ef)(p^a) = f(a)$ for every prime power $p^a$. We investigate the behaviour of $E^m\tau_k$, where $\tau_k$ is a…

We study principles of the form: if a name $\sigma$ is forced to have a certain property $\varphi$, then there is a ground model filter $g$ such that $\sigma^g$ satisfies $\varphi$. We prove a general correspondence connecting these name…

We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ as $x\to\infty$, which can be differentiated term by term…

Let $\sigma(x,\xi) $ be a sufficiently regular function defined on $R^d \times R^d.$ The pseudo-differential operator with symbol $\sigma$ is defined on the Schwartz class by the formula: \[f\to\sigma f(x)=\int_{R^d} \sigma(x,\xi)…

Bilinear Fourier multipliers of the form $e^{i (|\xi| + |\eta|+ |\xi + \eta|)} \sigma (\xi, \eta)$ are considered. It is proved that if $\sigma (\xi, \eta)$ is in the H\"ormander class $S^{m}_{1,0} (\mathbb{R}^{2n})$ with $m=-(n+1)/2$ then…

In 2000 Deaconescu raised a question whether there exists a composite $n$ for which $S_2(n)|\phi(n)-1$, where $\phi(n)$ is Euler's function and $S_2(n)$ is Schemmel's totient function. In this paper we prove that any such $n$ is odd,…

Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be odd is…

Given a sequence $\mathscr{A}=\{a_0<a_1<a_2\ldots\}\subseteq \mathbb{N}$, let $r_{\mathscr{A},h}(n)$ denote the number of ways $n$ can be written as the sum of $h$ elements of $\mathscr{A}$. Fixing $h\geq 2$, we show that if $f$ is a…

In this paper analyzes \textit{The Erd\H{o}s-Straus conjecture} asserts that $f$$(n)$ $>$ 0 for every $n$ $\geq$ 2, where $f(n)$ indicates the number of solutions to the Diophantine Equation…

We examine recursive monotonic functions on the Lindenbaum algebra of $\mathsf{EA}$. We prove that no such function sends every consistent $\varphi$ to a sentence with deductive strength strictly between $\varphi$ and…

For $n\leq 1.5 \cdot 10^{10}$, we have found a total number of 1268 solutions to the Erd\"os-Sierpi\'nski problem finding positive integer solutions of $\sigma(n)=\sigma(n+1)$, where $\sigma(n)$ is the sum of the positive divisors of n. On…

Let $\varphi$ be the Euler's function and fix an integer $k\ge 0$. We show that, for every initial value $x_1\ge 1$, the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi(x_n)+k$ for all $n\ge 1$ is eventually…

In this paper we introduce and study a family $\Phi_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k…

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(\sigma)=\sum_{n=1}^\infty X_nn^{-\sigma}$. We prove that the following holds almost surely…

Let the metric space $\mathbb R^n \setminus \sim$ be the metric space of $n$-sized unordered tuples of real numbers. In the following, it will be shown that if a function $\varphi: \mathbb R^m \to \mathbb R^n \setminus \sim$ is continuous,…