Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $λ_{A}(n) =(-1)^{Ω_A(n)}$ where $Ω_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville function, $A$ is the set of all primes. Denote $$L_A(n):=\sum_{k\leq n}λ_A(n)\quad{and}\quad R_A:=\lim_{n\to\infty}\frac{L_A(n)}{n}.$$ We show that for every $α\in[0,1]$ there is an $A\subset P$ such that $R_A=α$. Given certain restrictions on $A$, asymptotic estimates for $\sum_{k\leq n}λ_A(k)$ are also given. With further restrictions, more can be said. For {\em character--like functions} $λ_p$ ($λ_p$ agrees with a Dirichlet character $χ$ when $χ(n)\neq 0$) exact values and asymptotics are given; in particular $$\quad\sum_{k\leq n}λ_p(k)\ll \log n.$$ Within the course of discussion, the ratio $ϕ(n)/σ(n)$ is considered.
Euler wrote a formula expressing that l(n)/n is a completely multiplicative function with sum 0 (a CMO function) , where l(n) is the completely multiplicative function equal to -1 on the prime numbers (the Liouville function). We extend this formula to Beurling generalized numbers, using a result of Diamond. As an application we show how to construct a CMO function carried by a set of integers whose counting function is of the form D x^a (1+o(1) ) for any a between 0 and 1.
For completely multiplicative functions f(n) taking values 1 and -1, under certain conditions on f(n) we show that f(n) changes sign at least x exp(-7(log log x)sqrt(log x)) times as n runs through the integers <= x.
In this article we consider the completely multiplicative sequences $(a_n)_{n \in \mathbf{N}}$ defined on a field $\mathbf{K}$ and satisfying $$\sum_{p| p \leq n, a_p \neq 1, p \in \mathbf{P}}\frac{1}{p}<\infty,$$ where $\mathbf{P}$ is the set of prime numbers. We prove that if such sequences are automatic then they cannot have infinitely many prime numbers $p$ such that $a_{p}\neq 1$. Using this fact, we prove that if a completely multiplicative sequence $(a_n)_{n \in \mathbf{N}}$, vanishing or not, can be written in the form $a_n=b_nχ_n$ such that $(b_n)_{n \in \mathbf{N}}$ is a non ultimate…
In this article we prove that all completely multiplicative automatic sequences $(a_n)_{n \in \mathbf{N}}$ defined on $\mathbf{C}$, vanishing or not, can be written in the form $a_n=b_nχ_n$ where $(b_n)_{n \in \mathbf{N}}$ is an almost constant sequence, and $(χ_n)_{n \in \mathbf{N}}$ is a Dirichlet character.
In the first chapter, we will present a computation of the square value of the module of L functions associated to a Dirichlet character. This computation suggests to ask if a certain ring of arithmetic multiplicative functions exists and if it is unique. This search has led to the construction of that ring in chapter two. Finally, in the third chapter, we will present some propositions associated with this ring. The result below is one of the main results of this work : For F and G two completely multiplicative functions, $ s $ a complex number such as the dirichlet series $ D(F,s) $ and $ D(…
We establish several results concerning the expected general phenomenon that, given a multiplicative function $f:\mathbb{N}\to\mathbb{C}$, the values of $f(n)$ and $f(n+a)$ are "generally" independent unless $f$ is of a "special" form. First, we classify all bounded completely multiplicative functions having uniformly large gaps between its consecutive values. This implies the solution of the following folklore conjecture: for any completely multiplicative function $f:\mathbb{N}\to\mathbb{T}$ we have $$\liminf_{n\to\infty}|f(n+1)-f(n)|=0.$$ Second, we settle an old conjecture due to N.G. Chuda…
We prove that any $q$-automatic completely multiplicative function $f:\mathbb{N}\to\mathbb{C}$ essentially coincides with a Dirichlet character. This answers a question of J. P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative functions. Further, assuming two standard conjectures in number theory, the methods allows for removing the assumption of completeness.
We characterize all pairs of completely multiplicative functions $f,g:\mathbb{N}\to\mathbb{T}$ such that the orbit closure \[\overline{\{(f(n),g(n+1))\}_{n\ge 1}} \neq \mathbb{T}\times \mathbb{T}.\] In so doing, we settle an old conjecture of Zoltán Daróczy and Imre Kátai.
In this article, we study the behavior of consecutive values of random completely multiplicative functions $(X_n)_{n \geq 1}$ whose values are i.i.d. at primes. We prove that for $X_2$ uniform on the unit circle, or uniform on the set of roots of unity of a given order, and for fixed $k \geq 1$, $X_{n+1}, \dots, X_{n+k}$ are independent if $n$ is large enough. Moreover, with the same assumption, we prove the almost sure convergence of the empirical measure $N^{-1} \sum_{n=1}^N δ_{(X_{n+1}, \dots, X_{n+k})}$ when $N$ goes to infinity, with an estimate of the rate of convergence. At the end of t…
Let $f$ be a completely multiplicative function that assumes values inside the unit disc. We show that if $\sum_{n<x} f(n) \ll x/(\log x)^A$, $x>2$, for some $A>2$, then either $f(p)$ is small on average or $f$ pretends to be $μ(n)n^{it}$ for some $t$.
Consider a multiplicative function f(n) taking values on the unit circle. Is it possible that the partial sums of this function are bounded? We show that if we weaken the notion of multiplicativity so that f(pn)=f(p)f(n) for all primes p in some finite set P, then the answer is yes. We also discuss a result of Bronstein that shows that functions modified from characters at a finite number of places.
$CMO$ functions are completely multiplicative functions $f$ for which $\sum_{n=1}^\infty f(n)$ $=0$. These functions were first introduced and studied by Kahane and Saïas [5]. The main purpose of this paper is to generalise such functions to multiplicative functions and we shall call them $MO$ functions. More precisely, we define $MO$ functions to be multiplicative functions for which $\sum_{n=1}^\infty f(n) = 0$ and $\sum_{k=0}^\infty f(p^k)$ $\ne 0$ for all $p \in \mathbb{P}$. We give some properties and find examples of $MO$ functions, as well as pointing out the connection between these fu…
Let S be a subset of the unit disk, and let F(s) denote the class of completely multiplicative functions f such that f(p) is in S for all primes p. The authors' main concern is which numbers arise as mean-values of functions in F(s). More precisely, let Gamma_N(S) = {1/N sum_{n <= N} f(n): f in F(S)} and Gamma(S) = lim_{N -> infinity} Gamma_N(s). The authors call Gamma(S) the spectrum of the set S, and study its properties.
Bounded Exponential Sums with Multiplicative Coefficients
Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Cité, Sorbonne Université, CNRS（巴黎左岸数学研究所，巴黎西岱大学，索邦大学，法国国家科学研究中心）；Faculty of Mathematics, University of Cambridge（剑桥大学数学系）；School of Mathematics, University of Bristol（布里斯托尔大学数学学院）
We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.
In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-δ}$, for some $δ>0$, and if the Dirichlet series of $f$, say $F(s)$, is such that $F(1)=0$, then we obtain that for any $ε>0$, $\sum_{p\leq x}(1+f(p))\log p\ll x^{1-δ+ε}$. Moreover, a necessary condition for the existence of such $f$ is that the Riemann zeta function $ζ(s)$ has no zeros in the half plane $Re(s)>1-δ$.
We introduce several classes of monoids satisfying up to five axioms and establish basic theories on their arithmetics. The one satisfying all the axioms is named natural monoid. Two typical examples are 1) the monoid $\mathbb{N}$ of natural numbers in the group of positive rationals and 2) a certain monoid $\mathbb{S}$ in one of Thompson's groups. The latter one is non-abelian, which serves as an important example for non-commutative arithmetics. Defining primes in a non-abelian monoid $S$ is highly non-trivial, which relies on a concept we called `castling'. Three types of castlings are esse…
In this paper, we give a new proof and an extension of the following result of Bézivin. Let $f:\B{N}\to K$ be a multiplicative function taking values in a field $K$ of characteristic 0 and write $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ for its generating series. Suppose that $F(z)$ is algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $χ(n)$ such that $f(n)=n^k χ(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, $F(z)$ is either transcendental or rational. For $K=\B{C}$, we also prove that if $F(z)$ is a $D$-finite generating seri…
In this paper, we give various identities for the weighted average of the product of generalized Anderson-Apostol sums with weights concerning completely multiplicative function, completely additive function, logarithms, the Gamma function, Bernoulli polynomials and binomial coefficients respectively. Our results generalize many known results.
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 $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors.The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem 3). The same i…
We study sets of recurrence, in both measurable and topological settings, for actions of $(\mathbb{N},\times)$ and $(\mathbb{Q}^{>0},\times)$. In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form $\{(an+b)^{\ell}/(cn+d)^{\ell}: n \in \mathbb{N}\}$ are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.
We consider a sequence $\{f(p)\}_{p\ {\rm prime}}$ of independent random variables taking values $\pm 1$ with probability $1/2$, and extend $f$ to a multiplicative arithmetic function defined on the squarefree integers. We investigate upper bounds for $Ψ_f(x,y)$, the summatory function of $f$ on $y$-friable integers $\leq x$. We obtain estimations of the type $Ψ_f(x,y) \ll Ψ(x,y)^{1/2+ε}$, more precise formulas being given in suitable regions for $x,y$. In the special case $y=x$, this leads to the estimate $M_f(x) = \sum_{n \leq x} f(n) \ll \sqrt{x}\, (\log \log x)^{2+ε}$, which improves on pr…
We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an asymptotic upper bound of $O(\exp(-\exp( \frac{\log x}{C\log \log x })))$ on the exceptional probability that a particular truncation is negative, where $C$ is some positive constant.
We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely \[D_N(t) : = D_{f,\, N}(t)= \frac{1}{\sqrt{N}} \sum_{n\leqslant N} f(n) n^{it}, \] where $f$ is a completely multiplicative function with $|f(n)|=1$ for all $n\in\mathbb{N}$. We use Soundararajan's resonance method to produce large values of $\left|D_N(t)\right|$ uniformly for all such $f$. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds and only for almost all such $f$.
Let $f_1,\ldots,f_k : \mathbb{N} \rightarrow \mathbb{C}$ be multiplicative functions taking values in the closed unit disc. Using an analytic approach in the spirit of Halász' mean value theorem, we compute multidimensional averages of the shape $$x^{-l} \sum_{\mathbf{n} \in [x]^l} \prod_{1 \leq j \leq k} f_j(L_j(\mathbf{n}))$$ as $x \rightarrow \infty$, where $[x] := [1,x]$ and $L_1,\ldots, L_k$ are affine linear forms that satisfy some natural conditions. Our approach gives a new proof of a result of Frantzikinakis and Host that is distinct from theirs, with \emph{explicit} main and error te…
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}_0$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $α$, $f(p^α)<p^α$ and $f(p)\vert f(p^α)$. Suppose also that any prime that divides $f(p^α)$ also divides $pf(p)$. Define $f(0)=0$, and let $H(n)=\displaystyle{\lim_{m\rightarrow\infty}f^m(n)}$, where $f^m$ denotes the $m^{th}$ iterate of $f$. We prove that the function $H$ is completely multiplicative.
Turán 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 $f(n)=λ(n)$) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena. Firstly, we show that for every $\varepsilon>0$ there exists some $x_0\ge 1,$ such that for any completely multiplicative function $f$ satisfying $-1\le f(n)\le 1$, we have $$\sum_{n\le x}\frac{f(n)}{n}\ge -\frac{1}{(\log\log{x})^{1-\varepsilon}}, \quad x\ge x_0.$$ This improves a previous bound due to Granville and…
Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $μ$ be the Möbius function and $μ_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=μ^2g$ and $f=μ_2^2g$ have unbounded partial sums. Our proofs are built upon Klurman and Mangerel's proof of Chudakov's conjecture, Klurman's work on correlations of multiplicative functions and Tao's resolution of the Erdős discrepancy problem.
We study the problem of bounding the least prime that does not split completely in a number field. This is a generalization of the classic problem of bounding the least quadratic non-residue. Here, we present two distinct approaches to this problem. The first is by studying the behavior of the Dedekind zeta function of the number field near 1, and the second by relating the problem to questions involving multiplicative functions. We derive the best known bounds for this problem for all number fields with degree greater than 2. We also derive the best known upper bound for the residue of the De…
We consider the random functions $S_N(z):=\sum_{n=1}^N z(n) $, where $z(n)$ is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$. It is shown that ${\Bbb E} |S_N|\gg \sqrt{N}(\log N)^{-0.05616}$ and that $({\Bbb E} |S_N|^q)^{1/q}\gg_{q} \sqrt{N}(\log N)^{-0.07672}$ for all $q>0$.
We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs. As a first application, we show that for a completely multiplicative function $f : \mathbb{N} \to \{-1,1\},$ \begin{align*} \limsup_{x\to\infty}\Big|\sum_{n\leq x}μ^2(n)f(n)\Big|=\infty. \end{align*} This confirms a conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions. Secondly, we show that a completely multiplica…
Let $L(s)=\sum_{n=1}^{+\infty}\dfrac{a(n)}{n^s}$ be a Dirichlet series were $a(n)$ is a bounded completely multiplicative function. We prove that if $L(s)$ extends to a holomorphic function on the open half space $\Re s >1-δ$, $δ>0$ and $L(1)=0$ then such a half space is a zero free region of the Riemann zeta function $ζ(s)$. Similar results is proven for completely multiplicative functions defined on the space of the ideals of the ring of the algebraic integers of a number field of finite degree.
We call a set of positive integers closed under taking unitary divisors a unitary ideal. It can be regarded as a simplicial complex. Moreover, a multiplicative arithmetical function on such a set corresponds to a function on the simplicial complex with the property that the value on a face is the product of the values at the vertices of that face. We use this observation to solve the following problems: 1) Let r be a positive integer and c a real number. What is the maximum value that \sum_{s \in S}g(s) can obtain when S is a unitary ideal containing precisely r prime powers, and g is the mult…
Partition regularity in imaginary quadratic rings of integers
Universidad de Chile（智利大学）；Centro de Modelamiento Matemático（数学建模中心）；Complutense University of Madrid（马德里康普顿斯大学）；Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）；Virginia Tech（弗吉尼亚理工大学）
Let $n=\prod_p p^{ν_p(n)}$ denote the canonical factorization of $n\in \N$. The binomial convolution of arithmetical functions $f$ and $g$ is defined as $(f\circ g)(n)=\sum_{d\mid n} (\prod_p \binom{ν_p(n)}{ν_p(d)}) f(d)g(n/d),$ where $\binom{a}{b}$ is the binomial coefficient. We provide properties of the binomial convolution. We study the $\C$-algebra $({\cal A},+,\circ,\C)$, characterizations of completely multiplicative functions, Selberg multiplicative functions, exponential Dirichlet series, exponential generating functions and a generalized binomial convolution leading to various Möbius…
