Multiplicative Arithmetic Functions Research Articles

Multiplicative arithmetic functions and the generalized Ewens measure

Random integers, sampled uniformly from [1, x], share similarities with random permutations, sampled uniformly from Sn. These similarities include the Erdős–Kac theorem on the distribution of the number of prime factors of a random integer, and Billingsley’s theorem on the largest prime factors of a random integer. In this paper we extend this analogy to non-uniform distributions.Given a multiplicative function α: ℕ → ℝ≥0, one may associate with it a measure on the integers in [1, x], where n is sampled with probability proportional to the value α(n). Analogously, given a sequence {θi}i≥1 of non-negative reals, one may associate with it a measure on Sn that assigns to a permutation a probability proportional to a product of weights over the cycles of the permutation. This measure is known as the generalized Ewens measure.We study the case where the mean value of α over primes tends to some positive θ, as well as the weights α(p) ≈ (log p)γ. In both cases, we obtain results in the integer setting which are in agreement with those in the permutation setting.

A note on generating identities for multiplicative arithmetic functions

A note on generating identities for multiplicative arithmetic functions

A note on certain multiplicative arithmetic functions

We use the Apostol-Robbins theorem to determine for several arithmetic functions of interest in number theory

Mean square value of $L$-functions at $s=1$ for non-primitive characters, Dedekind sumsand bounds on relative class numbers

Let $d_0$ be a given square-free integer. We give an explicit formula $M_{d_0}(p) =A(d_0)(1+B_{d_0}(p)/p)$ for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the non-primitive odd Dirichlet characters modulo $d_0p$ induced by the odd Dirichlet characters modulo an odd prime $p$. Here $d_0\mapsto A(d_0)$ is an explicit multiplicative arithmetic function and $B_{d_0}(f)$ is a twisted sum over the divisors $d$ of $d_0$ of Dedekind sums $s(h,df)$. To prove that $f\mapsto B_{d_0}(f)$ is $d_0$-periodic, we find a new and closed formula for the Dedekind sums $f\mapsto s(a+bf,c+df)$, for fixed integers $a$, $b$, $c$ and $d$. We deduce explicit upper bounds for relative class numbers of cyclotomic number fields.

Some multiple Dirichlet series of completely multiplicative arithmetic functions

Let f_r: \mathbb{N}^r \longrightarrow \mathbb{C} be an arithmetic function of r variables, where r\geq 2. We study multiple Dirichlet series defined by \begin{equation*} D(f_r,s_1,\ldots,s_r)=\sum\limits_{\substack{n_1,\ldots,n_r=1 \\ (n_1,\ldots,n_r)=1}}^{+\infty}\frac{f_r(n_1,\ldots,n_r)}{n_1^{s_1}\cdots n_r^{s_r}}, \end{equation*} where f_r(n_1,\ldots,n_r)=f(n_1)\cdots f(n_r) and f is a completely multiplicative or a specially multiplicative arithmetic function of a single variable. We obtain formulas for these series expressed by infinite products over the primes. We also consider the cases of certain particular completely multiplicative and specially multiplicative functions.

On the number of $$k$$-compositions $$n$$ satisfying certain coprimality conditions

We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) on the number of $k$-compositions of $n$ satisfying some coprimality conditions. We substantially refine the error term concerning the number of $k$-compositions of $n$ with pairwise relatively prime summands. We use a different approach, based on properties of multiplicative arithmetic functions of $k$ variables and on an asymptotic formula for the restricted partition function.

Multiplicative functions in large arithmetic progressions and applications

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 $\tau(n-1)$ where $\tau$ denotes the divisor function.

Remarks on the Selberg–Delange method

Let $\varrho$ be a complex number and let $f$ be a multiplicative arithmetic function whose Dirichlet series takes the form $\zeta(s)^\varrho G(s)$, where $G$ is associated to a multiplicative function $g$. The classical Selberg-Delange method furnishes asymptotic estimates for averages of $f$ under assumptions of either analytic continuation for $G$, or absolute convergence of a finite number of derivatives of $G(s)$ at $s=1$. We consider different set of hypotheses, not directly comparable to the previous ones, and investigate how they can yield sharp asymptotic estimates for the averages of~$f$.

Multidimensional Schinzel-type theorems for multiplicative functions near prime tuples

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.

On some properties of a meta-Fibonacci sequence connected to Hofstadter sequence and Möbius function

Abstract The Hofstadter Q-sequence is perhaps the most known example of meta-Fibonacci sequence. Many authors have been interested in meta-Fibonacci sequences related to this sequence. For example, recently, it was studied by A. Alkan, N. Fox, and O. Aybar the connection between the Q-sequence and the Hofstadter-Conway $ 10,000 sequence. Here, in the similar spirit as their paper, we study the interplay between the Hofstadter Q-sequence and one of the more important multiplicative arithmetic function, namely, the Mobius function.

Riesz means of certain arithmetic functions

We give examples of completely multiplicative arithmetic functions that assume only the values ±1 and that have bounded first Cesàro means. The method of proof also yields some interesting identities involving special values of Dirichlet L-functions. In particular, we present some new class number formulas for quadratic fields.

On the least common multiple of several random integers

Let Ln(k) denote the least common multiple of k independent random integers uniformly chosen in {1,2,…,n}. In this article, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as n→∞ of f(Ln(k))nrk for a wide class of multiplicative arithmetic functions f with polynomial growth r∈R. Furthermore, we identify the limit as an infinite product of independent random variables indexed by the set of prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Erdős and Wintner (1939), Fernández and Fernández (2013) and Hilberdink and Tóth (2016).

Some properties and applications of non-trivial divisor functions

The jth divisor function d_j, which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative jth non-trivial divisor functionc_j, which counts the ordered factorisations of a positive integer into j factors each of which is greater than or equal to 2, is rather less well studied. Additionally, we consider the associated divisor functionc_j^{(r)}, for rge 0, whose definition is motivated by the sum-over divisors recurrence for d_j. We give an overview of properties of d_j, c_j and c_j^{(r)}, specifically regarding their Dirichlet series and generating functions as well as representations in terms of binomial coefficient sums and hypergeometric series. Noting general inequalities between the three types of divisor function, we then observe how their ratios can be expressed as binomial coefficient sums and hypergeometric series, and find explicit Dirichlet series and Euler products for some of these. As an illustrative application of the non-trivial and associated divisor functions, we show how they can be used to count principal reversible square matrices of the type considered by Ollerenshaw and Brée and so sum-and-distance systems of integers.

Weakly multiplicative arithmetic functions and the normal growth of groups

We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $\log$-uniformly continuous normal order is close to a function of the form $n\mapsto n^c$. As an application we show that a finitely generated, residually finite, infinite group, whose normal growth has a non-decreasing or a $\log$-uniformly continuous normal order is isomorphic to $(\mathbb{Z}, +)$.

Rigidity theorems for multiplicative functions

We establish several results concerning the expected general phenomenon that, given a multiplicative function $$f:\mathbb {N}\rightarrow \mathbb {C}$$ , 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 $$f:\mathbb {N}\rightarrow \mathbb {T}$$ we have $$\begin{aligned} \liminf _{n\rightarrow \infty }|f(n+1)-f(n)|=0. \end{aligned}$$ Second, we settle an old conjecture due to Chudakov (On the generalized characters. In: Actes du Congres International des Mathematiciens (Nice, 1970), Tome 1, p. 487. Gauthier-Villars, Paris) that states that any completely multiplicative function $$f:\mathbb {N}\rightarrow \mathbb {C}$$ that: (a) takes only finitely many values, (b) vanishes at only finitely many primes, and (c) has bounded discrepancy, is a Dirichlet character. This generalizes previous work of Tao on the Erdős Discrepancy Problem. Finally, we show that if many of the binary correlations of a 1-bounded multiplicative function are asymptotically equal to those of a Dirichlet character $$\chi $$ mod q then $$f(n) = \chi '(n)n^{it}$$ for all n, where $$\chi '$$ is a Dirichlet character modulo q and $$t \in \mathbb {R}$$ . This establishes a variant of a conjecture of H. Cohn for multiplicative arithmetic functions. The main ingredients include the work of Tao on logarithmic Elliott conjecture, correlation formulas for pretentious multiplicative functions developed earlier by the first author and Szemeredi’s theorem for long arithmetic progressions.

Mock characters and the Kronecker symbol

We introduce and study a family of functions we call the mock characters. These functions satisfy a number of interesting properties, and of all completely multiplicative arithmetic functions seem to come as close as possible to being Dirichlet characters. Along the way we prove a few new results concerning the behaviour of the Kronecker symbol.

Derivation of arithmetical functions under the Dirichlet convolution

We present the group-theoretic structure of the classes of multiplicative and firmly multiplicative arithmetical functions of several variables under the Dirichlet convolution, and we give characterizations of these two classes in terms of a derivation of arithmetical functions.

Moyennes effectives de fonctions multiplicatives complexes

We establish effective mean-value estimates for a wide class of multiplicative arithmetic functions, thereby providing (essentially optimal) quantitative versions of Wirsing’s classical estimates and extending those of Halász. Several applications are derived, including estimates for the difference of mean-values of so-called pretentious functions, local laws for the distribution of prime factors in an arbitrary set, and weighted distribution of additive functions.

Rational convolution roots of isobaric polynomials

In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under the convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first and second kind. Hence, we call them Stirling operators. In order to construct matrix representations of the roots of GFPs, we use Stirling operators of the first kind. We give explicit examples to show how Stirling operators of the second kind appear in low degree cases for the WIP-roots. As a consequence of the matrix construction we have matrix representations of multiplicative arithmetic functions under the Dirichlet product into its divisible closure.

On the average value of the least common multiple of k positive integers

We deduce an asymptotic formula with error term for the sum ∑n1,…,nk≤xf([n1,…,nk]), where [n1,…,nk] stands for the least common multiple of the positive integers n1,…,nk (k≥2) and f belongs to a large class of multiplicative arithmetic functions, including, among others, the functions f(n)=nr, φ(n)r, σ(n)r (r>−1 real), where φ is Euler's totient function and σ is the sum-of-divisors function. The proof is by elementary arguments, using the extension of the convolution method for arithmetic functions of several variables, starting with the observation that given a multiplicative function f, the function of k variables f([n1,…,nk]) is multiplicative.