Arithmetic Functions Research Articles

On a family of higher order recurrence relations: symmetries, formula solutions, periodicity and stability analysis

In this paper, we present formula solutions of a family of difference equations of higher order. We discuss the periodic nature of the solutions and we investigate the stability character of the equilibrium points. We utilize Lie symmetry analysis as part of our approach together with some number theoretic functions. Our findings generalize certain results in the literature.

The average value of a certain number-theoretic function over the primes

We consider functions $F:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}_{\geq 0}$ for which there exists a positive integer $n$ such that two conditions hold: $F(p)$ divides $n$ for every prime $p$, and for each divisor $d$ of $n$ and every prime $p$, we have that $d$ divides $F(p)$ iff $d$ divides $F(p \mod d)$. Following an approach of Khrennikov and Nilsson, we employ the prime number theorem for arithmetic progressions to derive an expression for the average value of such an $F$ over all primes $p$, recovering a theorem of these authors as a special case. As an application, we compute the average number of $r$-periodic points of a multivariate power map defined on a product $Z_{f_1(p)}\times\cdots\times Z_{f_m(p)}$ of cyclic groups, where $f_i(t)$ is a polynomial.

Objects generated by an arbitrary natural number. Part 4: New aspects

The set $\underline{SET}(n)$, generated by an arbitrary natural number $n$, was defined in [3]. There, and in [5,6], some arithmetic functions and arithmetic operators of a modal and topological types are defined over the elements of $\underline{SET}(n)$. Here, over the elements of $\underline{SET}(n)$ new arithmetic functions are defined and some of their basic properties are studied. Two standard modal topological structures over $\underline{SET}(n)$ are described. Perspectives for future research are discussed.

ADRA: Extending Digital Computing-In-Memory With Asymmetric Dual-Row-Activation

Computing in-memory (CiM) has emerged as an attractive technique to mitigate the von-Neumann bottleneck. Current digital CiM approaches for in-memory operands are based on multi-wordline assertion for computing bit-wise Boolean functions and arithmetic functions such as addition. However, most of these techniques, due to the many-to-one mapping of input vectors to bitline voltages, are limited to CiM of commutative functions, leaving out an important class of computations such as subtraction. In this paper, we propose a CiM approach, which solves the mapping problem through an asymmetric wordline biasing scheme, enabling (a) simultaneous single-cycle memory read and CiM of primitive Boolean functions (b) computation of any Boolean function and (c) CiM of non-commutative functions such as subtraction and comparison. While the proposed technique is technology-agnostic, we show its utility for ferroelectric transistor (FeFET)-based non-volatile memory. Compared to the standard near-memory methods (which require two full memory accesses per operation), we show that our method can achieve a full scale two-operand digital CiM using just one memory access, leading to a 23.2% -72.6% decrease in energy-delay product (EDP).

ON A CONJECTURE FOR A REFINEMENT OF THE SUM OF MINIMAL EXCLUDANTS

AbstractIn 2019, Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb.23(2) (2019), 249–254] introduced the arithmetic function $\sigma \textrm {mex}(n)$ , which denotes the sum of minimal excludants over all the partitions of n. Baruah et al. [‘A refinement of a result of Andrews and Newman on the sum of minimal excludants’, Ramanujan J., to appear] showed that the sum of minimal excludants over all the partitions of n is the same as the number of partition pairs of n into distinct parts. They proved three congruences modulo $4$ and $8$ for two functions appearing in this refinement and conjectured two further congruences modulo $8$ and $16$ . We confirm these two conjectures by using q-series manipulations and modular forms.

Unusual class of polynomials related to partitions

In this paper, we study, for a given arithmetic function f, the sequence of polynomials (P_{n}^{f}(t))_{n=0}^{infty }, defined by the recurrence P0f(x)=1,P1f(x)=x,Pnf(x)=xn∑k=1nf(k)Pn-kf(x),n≥2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} P_{0}^{f}(x)=1, &{} \\\\ P_{1}^{f}(x)=x, &{} \\\\ P_{n}^{f}(x)=\\frac{x}{n}\\sum _{k=1}^{n}f(k)P_{n-k}^{f}(x), &{} n\\ge 2. \\end{array}\\right. \\end{aligned}$$\\end{document}Using the ideas from the paper by Heim, Luca, and Neuhauser, we prove, under some assumptions on P_{n}^{f}(t) for 1le nle 10, that no root of unity can be a root of any polynomial P_{n}^{f}(t) for nin {mathbb {N}}. Then we specify the result to some functions f related to colored partitions.

On simply normal numbers with digit dependencies

AbstractGiven an integer and a set of prime numbers, the set of Toeplitz numbers comprises all elements of [0, b[ whose digits in the base‐b expansion satisfy for all and . Using a completely additive arithmetical function, we construct a number in that is simply Borel normal if, and only if, . We then provide an effective bound for the discrepancy.

Compositions of positive integers with 2sand 3s

AbstractIn this article, we consider compositions of positive integers with 2sand 3s. We see that these compositions lead us to results that involve Padovan numbers, and we give some tiling models of these compositions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling model’s method. Next, we obtain various identities of the compositions of positive integers with 2sand 3srelated to the Padovan numbers. The number of palindromic compositions of this type is determined, and some numerical arithmetic functions are defined. Finally, we provide a table that compares all of the results obtained from compositions of positive integers with 2sand 3s.

On arithmetic functions orthogonal to deterministic sequences

We prove the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property conjectured by Veech for Furstenberg systems of the Möbius function. This yields a combinatorial condition on the Möbius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the characterization persisting in the logarithmic setup. As a corollary, we obtain that the logarithmic Sarnak's conjecture holds if and only if the logarithmic Möbius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems.

New estimates for some integrals of functions defined over primes

In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function $\pi(x)$ and the Chebyshev $\vartheta$-function. Some of these estimates depend on the correctness of the Riemann hypothesis on the nontrivial zeros of the Riemann zeta function $\zeta(s)$.

Complete Bell Polynomials and Recurrence Relations for Arithmetic Functions

We use the Z-transform to solve a type of recurrence relation satisfied by the number of representations of an integer n as the sum of squares and as the sum of triangular numbers, and also by the color partitions of n; the corresponding solution is in terms of the complete Bell polynomials.

A note on certain multiplicative arithmetic functions

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

XOR-AND-XOR Logic Forms for Autosymmetric Functions and Applications to Quantum Computing

We propose a new three-level XOR-AND-XOR form for autosymmetric functions, called XORAX expression. In general, a Boolean function f over n variables is k-autosymmetric if it can be projected onto a smaller function fk, which depends on n-k variables only. We show that XORAX expressions can ease the reversible synthesis of autosymmetric functions, producing compact reversible networks, without inserting additional new input lines. Autosymmetry occurs especially for functions that exhibit a regular structure, as for instance arithmetic functions. For this reason, compact reversible networks for autosymmetric functions might be interesting for quantum computing. Experimental results validate the proposed approach.

On certain arithmetical functions of exponents in the factorization of integers

Some new results for the maximum and minimum exponents in factorizing integers are obtained. Related functions and generalized arithmetical functions are also introduced.

Systems engineering of cybersecured digital and information measuring systems based on the signature Boolean-polynomial algebra synthesis apparatus

Development of DIMSCC models with cyber-controllability of their apparatus and software for computing function (operations) in real time becomes the primary task since the problem relevance of the cyber security of digital information and measuring systems of control complexes (DIMSCC) for critical infrastructure objects is increasing. Based on an analysis of known digital systems functional control methods new term «functional cyber-controllability of the digital informational and measuring system» was defined in the article. New approach to operational functional control of hardware redundancy is proposed. This is the synthesis of DIMSCC computing operations. Hardware redundancy is a synthesized control node chosen for functional control of various DIMSCC basic arithmetic and logical functions (operations) which should be reduced to simple procedure of corresponding switching functionally complete combinational structures from a finite set (standard set). Concerning this, the synthesis task is as follows. Only inputs and outputs of specified object are available for functional control of arithmetic and logical operations on binary operands under external hacker influence. Combination type functional control node structure (operational compositional adaptation) should be developed depending on the type of controlled computing operation being performed. The result of the synthesis task solving is a system of formulae determination (signature control equations) of the signature functional control of all DIMSCC basic arithmetic and logical functions (operations) on a single basis of a single equivalent representation of their known formulaic expression by corresponding (adequate) descriptions in infix notation (infix models). Apparatus of Boolean-polynomial algebra is used aiming to realize infix notation. The practical advantage of the proposed approach to the synthesis of hardware redundancy of operational functional signature control of computing operations is the transition possibility from signature control equations formulae to their realization implementation in the form of a signature control node directly without additional interpreting and minimizing procedures use by simple logical composition (switching) functionally complete combinational structures from a finite set (standard set).

Arithmetic functions and fixed points of powers of permutations

Arithmetic functions and fixed points of powers of permutations

Mean value estimate of a hybrid arithmetic function attached to Fourier coefficients of Hecke–Maass forms

Mean value estimate of a hybrid arithmetic function attached to Fourier coefficients of Hecke–Maass forms

Some independence results related to finite trees.

We investigate some concrete independence results for systems of reverse mathematics which emerge from monotonicity properties of number-theoretic functions. Natural properties of the less than or equal to relation with respect to sums of natural numbers lead to independence results for first-order Peano arithmetic. Natural properties of the less than or equal to relation with respect to sums and products of natural numbers lead to independence results for arithmetical transfinite recursion. By considering number-theoretic functions of arbitrary arities, we obtain independence results for systems beyond arithmetical transfinite recursion. We discuss how these embeddability relations are related to tree embeddability relations and we consider variants where the tree embeddability relation is not assumed to preserve infima. The findings of this paper are complementary to results on Kruskal-like theorems proved earlier by the first author. This article is part of the theme issue 'Modern perspectives in Proof Theory'.

A Generalized Realizability and Intuitionistic Logic

Let V be a set of number-theoretical functions. We define a notion of V -realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of V -realizability if and only if some natural conditions for V hold.

Quotients of arithmetical functions under the Dirichlet convolution

We study existence of a solution of the arithmetical equation $f\ast h = g$ in $f,$ where $f\ast h$ is the Dirichlet convolution of arithmetical functions $f$ and $h,$ and derive an explicit expression for the solution. As applications we obtain expressions for the Möbius function $\mu$ and the so-called totients. As applications we also present our results on the arithmetical equation $f\ast h = g$ in the language of Cauchy convolution and further deconvolution in discrete linear systems.