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.
Roughly speaking, the spectrum of multiplicative functions is the set of all possible mean values. In this paper, we are interested in the spectra of multiplicative functions supported over powerful numbers. We prove that its real logarithmic spectrum takes values from $- \sqrt{2} / (4 + \sqrt{2}) = -0.26160...$ to $1$ while it is known that the logarithmic spectrum of real multiplicative functions over all natural numbers takes values from $0$ to $1$. In the course of this study, we correct a proof of Granville and Soundararajan concerning contribution of small primes in the study of mean val…
We obtain an upper bound for the multiplicative energy of the spectrum of an arbitrary set from $\mathbb{F}_p$, which is the best possible up to the results on exponential sums over subgroups.
We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational leading coefficient and for each multiplicative function $\bnu:\N\to\C$, $|\bnu|\leq1$, we have\[ \frac{1}{M} \sum\_{M\le m\textless{}2M} \frac{1}{H} \left| \sum\_{m\le n \textless{} m+H} e^{2πiP(n)}\bnu(n) \right|\longrightarrow 0 \] as $M\to\infty$, $H\to\infty$, $H/M\to 0$.
We study arithmetic inequalities for multiplicative, sub(super)-multiplicative, sub(super)-homogeneous functions. Applications for the classical arithmetic functions are pointed out.
A characterization of multiplicative (and additive) arithmetical functions is given. Using this characterization, we show that the group of multiplicative arithmetical functions is isomorphic to the group of additive arithmetical functions.
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 bound for $E(Ω,g,x)$.
In this paper we study the mean values of some multiplicative functions connected with the divisor function on the short interval of summation. The asymptocic values for such mean values are proved.
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
We consider the classes of quasimultiplicative, semimultiplicative and Selberg multiplicative functions as extensions of the class of multiplicative functions. We apply these concepts to Ramanujan's sum and its analogue with respect to regular integers (mod r).
A method of estimating sums of multiplicative functions braided with Dirichlet characters is demonstrated, leading to a taxonomy of the characters for which such sums are large.
Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.
Given a multiplicative function f satisfying |f(n)| <= 1 for all n, the authors study the problem of obtaining explicit upper bounds on the mean-value 1/x |sum_{n <= x} f(n)|.
We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.
If a multiplicative function $f$ satisfies $f(a^2+b^2+c^2) = f(a)^2+f(b)^2+f(c)^2$ for all positive integers $a$, $b$, and $c$, then $f$ is an identity function.
$CMO$ 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…
We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao's method for constructing elements in the finite field $\mathbb{F}_{q^n}$ whose orders are larger than any polynomial in $n$ when $n$ becomes large. Additionally, we discuss the finiteness of polynomials which translate a given finite set of polynomials to become multiplicatively dependent.
Let $k \geq 3$. If a multiplicative function $f$ satisfies \[ f(a_1^3 + a_2^3 + \cdots + a_k^3) = f(a_1^3) + f(a_2^3) + \cdots + f(a_k^3) \] for all $a_1, a_2, \ldots, a_k \in \mathbb{N}$, then $f$ is the identity function. The set of positive cubes is said to be a $k$-additive uniqueness set for multiplicative functions. But, the condition for $k=2$ can be satisfied by infinitely many multiplicative functions. Besides, if $k \geq 3$ and a multiplicative function $g$ satisfies \[ g(a_1^3 + a_2^3 + \cdots + a_k^3) = g(a_1)^3 + g(a_2)^3 + \cdots + g(a_k)^3 \] for all $a_1, a_2, \ldots, a_k \in \…
We provide examples of multiplicative functions $f$ supported on the squarefree integers, such that on primes $f(p)=\pm1$ and such that $M_f(x):=\sum_{n\leq x} f(n)=o(\sqrt{x})$. Further, by assuming the Riemann hypothesis (RH) we can go beyond $\sqrt{x}$-cancellation.
In this paper, we investigate the asymptotics of a class of weighted sums over multiplicative functions and apply our results to deduce a stronger asymptotic form of Yitang Zhang's smoothened GPY sieve with coefficient expressions that are friendly to numerical computations.
We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors' new proof of Halasz's theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields. We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.
Let $K$ be a field of characteristic 0, $f:\mathbb{N}\to K$ be a multiplicative function, and $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ be 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, the generating function of a multiplicative function $f:\mathbb{N}\to K$ is either transcendental or rational.
Assuming Dickson's conjecture, we obtain multidimensional analogues of recent results on the behavior of certain multiplicative arithmetic functions near twin-prime arguments. This is inspired by analogous unconditional theorems of Schinzel undertaken without primality assumptions.
We show that smooth-supported multiplicative functions $f$ are well-distributed in arithmetic progressions $a_1a_2^{-1} \pmod q$ on average over moduli $q\leq x^{3/5-\varepsilon}$ with $(q,a_1a_2)=1$.
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.
This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression and to the Fourier coefficients of automorphic cusp forms.
Fix $k \ge 3$. If a multiplicative function $f$ satisfies \[ f(x_1+x_2+\dots+x_k) = f(x_1) + f(x_2) + \dots + f(x_k) \] for arbitrary positive triangular numbers $x_1, x_2, \dots, x_k$, then $f$ is the identity function. This extends Chung and Phong's work for $k=2$.
We adapt (over $\mathbb{F}_2$) the general notions of multiplicative function, Dirichlet convolution and Inverse. We get some interesting results, namely necessary conditions for an odd binary polynomial to be perfect. Note that we are inspired by the "analogous" works in \cite{Gall-Rahav-newcongr} and \cite{Touchard}, about odd perfect numbers.
We study sums with multiplicative functions that take values over a non-homogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas such as the number of integers in a Beatty sequence representable as a sum of two squares up to a given magnitude.
We establish an interesting upper bound for the moments of truncated Dirichlet convolution of Möbius function, a function noted $M(n,z)$. Our result implies that $M(n,j)$ is usually quite small for $j \in \{1,\dots,n\}$. Also, we establish an estimate for the multiplicative energy of the set of divisors of an integer $n$.
In this paper, we study the average size of the product of a number and its multiplicative inverse modulo a prime p. This turns out to be related to moments of L-functions and leads to a curious asymptotic formula for a certain triple exponential sum.
Let $f$ and $g$ be $1$-bounded multiplicative functions for which $f*g=1_{.=1}$. The Bombieri-Vinogradov Theorem holds for both $f$ and $g$ if and only if the Siegel-Walfisz criterion holds for both $f$ and $g$, and the Bombieri-Vinogradov Theorem holds for $f$ restricted to the primes.
Let $\mathtt{PRIMES}$ be the set of all primes. We show that a multiplicative function which satisfies \[ f(p+q-2) = f(p) + f(q) - f(2) \text{ for }p,q \in \mathtt{PRIMES} \] is one of the following: \begin{enumerate} \item $f$ is the identity function \item $f$ is the constant function with $f(n)=1$ \item $f(n)=0$ for $n \ge2$ unless $n$ is odd and squareful. \end{enumerate} As a consequence, a multiplicative function which satisfies \[ f(a+b) = f(a) + f(b) \text{ for }a,b \in \mathtt{PRIMES}-1 \] is the identity function.
P. V. Chung showed that there are many multiplicative functions $f$ which satisfy $f(m^2+n^2) = f(m^2)+f(n^2)$ for all positive integers $m$ and $n$. In this article, we show that if more than $2$ squares in the additive condition are involved, then such $f$ is uniquely determined. That is, if a multiplicative function $f$ satisfies \[ f(a_1^2 + a_2^2 + \dotsb + a_k^2) = f(a_1^2) + f(a_2^2) + \dotsb + f(a_k^2) \] for arbitrary positive integers $a_i$, then $f$ is the identity function. In this sense, we call the set of all posotive squares a \emph{$k$-additive uniqueness set} for multiplicativ…
Let $k$ be an integer greater than or equal $4$. We show that if a multiplicative function $f$ satisfies \[ f(x_1^2 + x_2^2 + \dots + x_k^2) = f(x_1)^2 + f(x_2)^2 + \dots + f(x_k)^2 \] for all positive integers $x_i$'s, then $f$ is the identity function.
We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$-divisor functions, where $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Halász's Theorem.
We use multiple zeta functions to prove, under suitable assumptions, precise asymptotic formulas for the averages of multivariable multiplicative functions. As applications, we prove some conjectures on the average number of cyclic subgroups of the group ${\mathbb Z}_{m_1}\times\dots\times {\mathbb Z}_{m_n}$ and multivariable averages associated with the LCM function.
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
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.
A celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalizes a well known convergence result of Wirsing. The key ingredients are a recent structural result for bounded multiplicative functions proved by the authors and…
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.
Let $f$ be a multiplicative function which satisfies \[ f(a^2+b^2+c^2+d^2) = f(a^2+b^2)+f(c^2+d^2) \] for positive integers $a$, $b$, $c$, and $d$. We show that $f$ is the identity function provided that $f(3)\,f(11) \ne 0$. Otherwise, $f(n)=0$ for all $n \ge 2$ except for $n=3,9,11$.
Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limita…
Under fairly general conditions, we show that families of integer-valued polynomial-like multiplicative functions are uniformly distributed in coprime residue classes mod $p$, where $p$ is a growing prime (or nearly prime) modulus. This can be seen as complementary to work of Narkiewicz, who obtained comprehensive results for fixed moduli.
We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erd{\H o}s-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by $τ(n-1)$ where $τ$ denotes the divisor function.
Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result $$\max_{n\leq x} \log d(n) = \frac{\log x}{\log \log x} (\log 2 + o(1)). $$ On the contrary, for many multiplicative functions, the maximal order of iterations of the functions remains widely open. The case of the iterated divisor function was only solved recently, answering a question of Ramanujan from 1915. Here we determine the maximal order of $\log f(f(n))$ for a class of multiplicative fun…
If a multiplicative function $f$ is commutable with a quadratic form $x^2+xy+y^2$, i.e., \[ f(x^2+xy+y^2) = f(x)^2 + f(x)\,f(y) + f(y)^2, \] then $f$ is the identity function. In other hand, if $f$ is commutable with a quadratic form $x^2-xy+y^2$, then $f$ is one of three kinds of functions: the identity function, the constant function, and an indicator function for $\mathbb{N}\setminus p\mathbb{N}$ with a prime $p$.
Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe significantly analogous results on the set \[ {\mathcal L}(α)=\left\{\left.\limsup_{n\to \infty}\|ξα^n\|\ \right|\ ξ\in {\mathbb R}\right\}, \] where $\|x\|$ is the distance from $x$ to the nearest integer. First, we show that ${\mathcal L}(α)$ is closed in $[0,1/2]$ for any Pisot number $α$. Then we consider the case where $α$ is an integer with $α\geq 2$, or a quadratic unit with $α\ge 3$. We show that ${\mathcal …
The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading factors of the infinite product over zeta-functions. If rooted at the Dirichlet series for powers, for sums-of-divisors and for Euler's totient, the inheritance of multiplicativity through Dirichlet convolution or ordinary multiplication of pairs of arithmetic functions generates most of the results.
We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant function $1$ and that for some prime $q$, $\sum_{k=0}^\infty \frac{f(q^k)}{q^k}=0$. These combined with other conditions guarantee that these functions are periodic and have sum equal to zero inside each period. Further, we study the class of multiplicative functions $f=f_1\ast f_2$, where each $f_j$ is multiplicative and periodic with bounded partial sums.…
We give an asymptotic formula for correlations \[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are bounded "pretentious" multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences:\ First, we characterize all multiplicative functions $f:\mathbb{N}\to\{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{R…
We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic $L$-function on $GL_m (m \geq 2)$ and polynomial nilsequences has logarithmic decay; the other is that the mean value of the Möbius function, coefficients of automorphic $L$-function and polynomial nilsequences also has logarithmic decay.
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…
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…
