Showing papers by "László Tóth published in 2020"

On certain sums concerning the gcd’s and lcm’s of k positive integers

Abstract: We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd’s and lcm’s of k positive integers, where k ≥ 2 is fixed. We refine and generalize an asymptotic formula of Bordell`es (2007), and extend certain related results of Hilberdink and T´oth (2016). We also formulate some conjectures and open problems.

Menon-type identities again: A note on a paper by Li, Kim and Qiao

Abstract: We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple character free approach for the proof.

Coefficients of (inverse) unitary cyclotomic polynomials

Abstract: The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $\Phi_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime factors using numerical semigroups, respectively Bachman's inclusion-exclusion polynomials. Given $m \ge 1$ we show that every integer occurs as a coefficient of $\Phi^*_{mn}(x)$ for some $n\ge 1$ following Ji, Li and Moree [9]. Here $n$ will typically have many different prime factors. We also consider similar questions for the polynomials $(x^n-1)/\Phi_n^*(x)$, the inverse unitary cyclotomic polynomials.

Menon-type identities concerning additive characters

Abstract: By considering even functions ( $$\hbox {mod}\ n$$ ), we generalize a recent Menon-type identity by Li and Kim, involving additive characters of the group $${\mathbb {Z}}_n$$ . We use a different approach, based on certain convolutional identities. Some other applications, including related formulas for Ramanujan sums, are discussed as well.

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

Abstract: 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.

Unitary cyclotomic polynomials

Abstract: The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials. We formulate some basic properties of unitary cyclotomic polynomials and study how they are connected with cyclotomic, inclusion-exclusion and Kronecker polynomials. Further, we derive some related arithmetic function identities involving the unitary analog of the Dirichlet convolution.

Transcendental Infinite Products Associated with the $\pm 1$ Thue-Morse Sequence

Abstract: Infinite products associated with the $\pm 1$ Thue-Morse sequence whose value is rational or algebraic irrational have been studied by several authors. In this short note we prove three new infinite product identities involving ${\pi}$, $\sqrt{2}$, and the $\pm 1$ Thue-Morse sequence, building on a result by Allouche, Riasat, and Shallit. We then use our method to find a new expression for a product appearing within the context of the Flajolet-Martin constant.

Another generalization of Euler's arithmetic function and of Menon's identity

Abstract: We define the $k$-dimensional generalized Euler function $\varphi_k(n)$ as the number of ordered $k$-tuples $(a_1,\ldots,a_k)\in {\Bbb N}^k$ such that $1\le a_1,\ldots,a_k\le n$ and both the product $a_1\cdots a_k$ and the sum $a_1+\cdots +a_k$ are prime to $n$ We investigate some of properties of the function $\varphi_k(n)$, and obtain a corresponding Menon-type identity

On certain sums of arithmetic functions involving the gcd and lcm of two positive integers

Abstract: We obtain asymptotic formulas with remainder terms for the hyperbolic summations $\sum_{mn\le x} f((m,n))$ and $\sum_{mn\le x} f([m,n])$, where $f$ belongs to certain classes of arithmetic functions, $(m,n)$ and $[m,n]$ denoting the gcd and lcm of the integers $m,n$. In particular, we investigate the functions $f(n)=\tau(n), \log n, \omega(n)$ and $\Omega(n)$. We also define a common generalization of the latter three functions, and prove a corresponding result.

On the average number of cyclic subgroups of the groups $${\mathbb {Z}}_{n_1} \times {\mathbb {Z}}_{n_2}\times {\mathbb {Z}}_{n_3}$$Zn1×Zn2×Zn3 with $$n_1,n_2,n_3\le x$$n1,n2,n3≤x

Abstract: Let $${{\mathbb {Z}}}_{n}$$ be the additive group of residue classes modulo n. Let $$c(n_1,n_2,n_3)$$ denote the number of cyclic subgroups of the group $${{\mathbb {Z}}}_{n_1}\times {{\mathbb {Z}}}_{n_2}\times {{\mathbb {Z}}}_{n_3}$$, where $$n_1, n_2$$ and $$n_3$$ are arbitrary positive integers. In this paper we obtain an asymptotic formula for the sum $$\sum _{n_1,n_2,n_3\le _x} c(n_1,n_2,n_3).$$

On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers.

Abstract: In this paper we study the $b$-ary expansions of the square roots of the function defined by the recurrence $f_b(n)=b f_b(n-1)+n$ with initial value $f(0)=0$ taken at odd positive integers $n$, of which the special case $b=10$ is often referred to as the "schizophrenic" or "mock-rational" numbers. Defined by Darling in $2004$ and studied in more detail by Brown in $2009$, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases $b\geq2$ by formally defining the schizophrenic pattern present in the $b$-ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within.