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 $σ_A(n)= \sum_{d\in A(n)} d$, where A(n) is a subset of the set of all positive divisors of $n$, including the divisor-sum function $σ(n)$ and its unitary and exponential analogues. We also give the minimal order of a new class of Euler-type functions, including the Euler-function $ϕ(n)$ and its unitary analogue.
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.
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…
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.
Let $F(z)$ be a $k$-regular series in $\mathbb{Z}[[z]]$ and $b$ be an integer with $b\ge2$. Bell, Bugeaud and Coons [BelBC] proved that $F(\frac{1}{b})$ is either rational or transcendental. In [Mi], we introduce a generalized $k$-regular sequence as a unification of several kinds of important sequences including $k$-regular, $k$-additive and $k$-multiplicative sequences. In this paper, we give a generalization of the result of Bell, Bugeaud and Coons for certain generalized $k$-regular series. Especially, we show that the values of irrational generating functions of certain sum of $k$-additiv…
We describe the points of diff erentiability of the multiplicative autocorrelation function of the "fractional part" function. In connection with this question, we study series involving the fi rst Bernoulli function, the arithmetical function "number of divisors", and the Gauss map from the theory of continued fractions. A key role is played by a function defi ned in 1933 by J. R. Wilton, similar to the Brjuno function of dynamical systems theory. A unifying theme of our exposition is the use of functional equations involving the Gauss map, allowing us to reprove and re fine a theorem of Wilt…
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…
Extending the partition function multiplicatively to a function on partitions, we show that it has a unique maximum at an explicitly given partition for any $n\neq 7$. The basis for this is an inequality for the partition function which seems not to have been noticed before.
We prove that strongly $b$-multiplicative functions of modulus $1$ along squares are asymptotically orthogonal to the Möbius function. This provides examples of sequences having maximal entropy and satisfying this property.
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…
Let $\gcd(k,j)$ be the greatest common divisor of the integers $k$ and $j$. For any arithmetical function $f$, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is $$\sum_{k\leq x}\frac{1}{k} \sum_{j=1}^{k}f(\gcd(k,j)) \log j.$$ More precisely, we give asymptotic formulas for various multiplicative functions such as $f=id$, $ϕ$, $id_{1+a}$ and $ϕ_{1+a}$ with $-1<a<0$. We also establish some formulas of Dirichlet series having coefficients of the sum function $\sum_{j=1}^{k}s_{k}(j)\log j$ where $s_{k}(j)$ is Anderson--Ap…
In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$ \sum_{k\leq x}\frac{1}{k^s} \sum_{j=1}^{k^s}\logΓ\left(\frac{j}{k^s}\right) \sum_{\substack{d|k \\ d^{s}|j}}f*μ(d) \quad {\rm and } \quad \sum_{k\leq x}\frac{1}{k^s}\sum_{j=0}^{k^{s}-1} B_{m}\sum_{\substack{d|k \\ d^{s}|j}} f*μ(d) $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We give asymptotic formulas for them with various multiplicative functions $f$. We also consider several formulas of the Dirichlet series associated with the above identities. Th…
For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = \frac1{\sqrt{N}} \sum_{n \leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $\varepsilon>0$ we show that, with high probability, $$ \exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp( (\log N)^{1/2+\varepsilon}). $$
Let $\taue_k \colon \Z\to\Z$ be a multiplicative function such that $ \taue_k(p^a) = \sum_{d_1... d_k=a} 1 $. In the present paper we introduce generalizations of $\taue_k$ over the ring of Gaussian integers $\Zi$. We determine their maximal orders by proving a general result and establish asymptotic formulas for their average orders.
Let $F$ be a non-Archimedean local field, $A$ be a central simple $F$-algebra, and $G$ be the multiplicative group of $A$. It is known that for every irreducible supercuspidal representation $π$, there exists a $[G, π]_{G}$-type $(J, λ)$, called a (maximal) simple type. We will show that $[G, π]_{G}$-types defined over some maximal compact subgroup are unique up to $G$-conjugations under some unramifiedness assumption on a simple stratum.
We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1 \leq q \leq \frac{c\log x}{\log\log x}$. In the Steinhaus case, we show that $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q} = e^{O(q^2)} x^q (\frac{\log x}{q\log(2q)})^{(q-1)^2}$ on this whole range. In the Rademacher case, we find a transition in the behaviour of the moments when $q \approx (1+\sqrt{5})/2$, where the size starts to be dominated by "orthogonal" rather than "unitary" behaviour. We als…
The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ for which $\lim_{r\to 1}(1-r)^{1/q}\log^+M_{\infty}(r,f)=0,$ where $M_{\infty}(r,f)=\max_{\vert z\vert\le r}\vert f(z)\vert$ with $0<r<1$. Stoll [5, Theorem 3.2] proved that the space $F^p$ with the topology given by the family of seminorms $\left\{\Vert \cdot\Vert_{q,c}\right\}_{c>0}$ defined for $f\in F^q$ as $\Vert f\Vert_{q,c}:=\sum_{n=0}^{\infty}\vert a_n\vert\exp\left(-cn^{1/(q+1)} \right)<\infty,$ is a countably normed Fréchet algebra. Notice that for each $p>1$, $F^p$ is the Fréchet …
