Related papers: On multiplicative functions with small partial sum…

Sign changes in sums of arithmetic functions and their inverses are a subtle topic with room to grow new results. Suppose that $S_f(x) := \sum_{n \leq x} f(n)$ is the summatory function of some arithmetic function $f$ such that $f(1) \neq…

A positive integer $n$ is practical if every $m \leq n$ can be written as a sum of distinct divisors of $n$. One can generalize the concept of practical numbers by applying an arithmetic function $f$ to each of the divisors of $n$ and…

For an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ on a neighbourhood of a closed disc $D\subset {\bf C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the…

The arithmetic partial derivative (with respect to a prime $p$) is a function from the set of integers that sends $p$ to 1 and satisfies the Leibniz rule. In this paper, we prove that the $p$-adic valuation of the sequence of higher order…

Let $g_0,\dots,g_k: {\bf N} \to {\bf D}$ be $1$-bounded multiplicative functions, and let $h_0,\dots,h_k \in {\bf Z}$ be shifts. We consider correlation sequences $f: {\bf N} \to {\bf Z}$ of the form $$ f(a):= \widetilde{\lim}_{m \to…

For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.

Consider a normal function $f$ on the ordinals (i. e. a function $f$ that is strictly increasing and continuous at limit stages). By enumerating the fixed points of $f$ we obtain a faster normal function $f'$, called the derivative of $f$.…

The aim of this work is to illustrate a conditional result involving the exponential sums over primes in short intervals under the assumption that both the Generalized Riemann Hypothesis and the Density Hypothesis for Dirichlet…

Applying Baaz's Generalization Method and a new technique to, respectively, proofs and denumerable simple graphs, diverse arithmetical patterns are observed. In particular, sufficient conditions for a number to be a divisor of a Fermat…

We define "splitting functions of level l" for any integer l>0. These functions generalize Dwork's splitting functions : they allow us to represent additive characters of order $p^l$. Then we use these functions to obtain a Stickleberger…

For set $A\subset {\mathbb {F}_p}^*$ define by ${\mathsf{sf}}(A)$ the size of the largest sum--free subset of $A.$ Alon and Kleitman showed that ${\mathsf{sf}} (A) \ge |A|/3+O(|A|/p).$ We prove that if ${\mathsf{sf}} (A)-|A|/3$ is small…

We provide uniform bounds on mean values of multiplicative functions under very general hypotheses, detecting certain power savings missed in known results in the literature. As an application, we study the distribution of the…

We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's…

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series $D(\alpha;z)=\sum_{n\geq 2}(\log n)^{\alpha}(\eta_n+{\rm i} \theta_n)/n^z$, properly scaled and normalized, where…

Under the formalism of annealed averaging of the partition function, a type of random multifractal measures with their multipliers satisfying exponentially distributed is investigated in detail. Branching emerges in the curve of generalized…

Let $k \ge 2$ be a fixed integer. We define the multiplicative function $D_k(n) = d_k(n)/d_k^*(n)$, such that $d_k(n)$ is the Piltz divisor function and $d_k^*(n) = k^{\omega(n)}$ is its unitary analogue, where $\omega(n)$ is the number of…

Let $p$ be a prime number, $V$ a discrete valuation ring of unequal caracteristics $(0,p)$, $G$ a smooth affine algebraic group over $Spec \,V$. Using partial divided powers techniques of Berthelot, we construct arithmetic distribution…

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}_0$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $\alpha$, $f(p^{\alpha})<p^{\alpha}$ and $f(p)\vert f(p^{\alpha})$. Suppose also that any prime that…

In the framework of a strictly local regular Dirichlet space ${\bf X}$ we introduce the subspaces $PW_{\omega},\>\>\omega>0,$ of Paley-Wiener functions of bandwidth $\omega$. It is shown that every function in $PW_{\omega},\>\>\omega>0,$ is…

We make progress on a conjecture of Cilleruelo on the growth of the least common multiple of consecutive values of an irreducible polynomial $f$ on the additional hypothesis that the polynomial be even. This strengthens earlier work of…