Arithmetic Functions Research Articles

Reducing round-off errors using pair-precision technique and its application to the orbit integration

The pair-precision technique, used to reduce round-off errors, was applied to the 6th-order symplectic integration method for integrating the equation of motion of the two-body problem. Analytical comparisons confirm that the implementation of pair-precision arithmetic operation functions reduces integration errors by a factor of 108 and a precision gain of around 20 digits along the orbit is obtained. Manipulated integration comes at the cost of approximately a 3.9-fold increase in CPU time, however, it is 2.1 times more efficient than using quadruple-precision computation. Integration with the pair-precision technique can be used for checking the validity and setting time limits for similar numerical calculations. More accurate calculations can extend the usefulness of predictions for evolutionary studies in the dynamics of the Solar System.

NEW ARITHMETIC FUNCTION RELATED TO THE LEAST COMMON MULTIPLE

The author’s last two papers concerned the study of arithmetic functions related to the greatest common divisor. In the present paper, similarly to the two previous papers we will define new multiplicative arithmetic functionsrelated to the least common multiple and we will study several interesting properties of them. Also, we will discuss the values of two special functionsof them at perfect numbers.

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.

Double Dirichlet series associated with arithmetic functions II

This paper is a continuation of our previous work on double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the Möbius function, and so on. We consider the analytic behaviour around the non-positive integer points on singularity sets which are points of indeterminacy. In particular, we show a certain reciprocity law of their residues. Also on this occasion we correct some inaccuracies in our previous paper.

Objects generated by an arbitrary natural number. Part 3: Standard modal-topological aspect

The set $\underline{SET}(n)$ generated by an arbitrary natural number $n$, was defined in [3]. There, and in [4], some arithmetic functions and arithmetic operators of a modal type are defined over the elements of $\underline{SET}(n)$. Here, over the elements of $\underline{SET}(n)$ arithmetic operators of a topological type are defined and some of their basic properties are studied. Perspectives for future research are discussed.

On certain equations and inequalities involving the arithmetical functions φ(n) and d(n) – II

In papers [3] and [5] we have studied certain equations and inequalities involving the arithmetical functions \varphi(n) and d(n). In this paper we will consider some other equations. Some open problems will be stated, too.

On a sum involving certain arithmetic functions on Piatetski-Shapiro and Beatty sequences

On a sum involving certain arithmetic functions on Piatetski-Shapiro and Beatty sequences

Association Between Body Mass Index and Cognitive Function Among Older Adults in India: Findings from a Cross-Sectional Study

ABSTRACT Objective The study aimed to explore the association between body mass index (BMI) and cognitive function among older adults in India. Methods Cross-sectional data on 21,000 older adults aged 60 years and older from the first wave of the Longitudinal Ageing Study of India (LASI), conducted in India during 2017-18, were used. Bivariate analysis and multivariate linear regression models were applied. The cognition tests include memory, orientation, arithmetic function, executive function, and object naming. Results The mean overall cognition score was 24.6 in the study population. Further, BMI status showed a significant and positive association with cognition. The association of underweight with poor cognition (β=-0.72; 95% CI = −0.89, −0.54) whereas overweight (β = 0.57; 95% CI = 0.39, 0.75) and obese (β = 0.97; 95% CI = 0.68, 1.26) with better cognition remained statistically significant after adjusting for sociodemographic, health-related behavior, and health covariates. The mean cognition score of female older adults was always lower than male older adults with normal BMI, irrespective of their BMI categories. Conclusions This study shows that an underweight BMI is associated with poor cognition in both male and female older adults. In the context of providing health care for older individuals, underweight individuals should be given more attention in India.

On a sum involving small arithmetic function and the integral part function

Let f be a “small” arithmetic function in the sense that f=g⁎1 and g(n)≪n−j, where j is a fixed non-negative number. In this paper, we study the sum∑n⩽xf([x/n])[x/n]k as x→∞, where [⋅] denotes the integral part function and k is a fixed non-negative number. Our results generalize the very recent work of Stucky, also combine and generalize the original two types of sums studied by Bordellès-Dai-Heyman-Pan-Shparlinski.

Computing Arithmetic Functions Using Immobilised Enzymatic Reaction Networks.

Living systems use enzymatic reaction networks to process biochemical information and make decisions in response to external or internal stimuli. Herein, we present a modular and reusable platform for molecular information processing using enzymes immobilised in hydrogel beads and compartmentalised in a continuous stirred tank reactor. We demonstrate how this setup allows us to perform simple arithmetic operations, such as addition, subtraction and multiplication, using various concentrations of substrates or inhibitors as inputs and the production of a fluorescent molecule as the readout.

Computing Arithmetic Functions Using Immobilised Enzymatic Reaction Networks

AbstractLiving systems use enzymatic reaction networks to process biochemical information and make decisions in response to external or internal stimuli. Herein, we present a modular and reusable platform for molecular information processing using enzymes immobilised in hydrogel beads and compartmentalised in a continuous stirred tank reactor. We demonstrate how this setup allows us to perform simple arithmetic operations, such as addition, subtraction and multiplication, using various concentrations of substrates or inhibitors as inputs and the production of a fluorescent molecule as the readout.

Regular Elements and Von-Neumann Inverses of a Class of Zero Symmetric Local Near-Rings Admitting Frobenius Derivations

Let N be a zero-symmetric local near-ring. An element \({x}\) \(\in\) N is either regular, zero or a zero divisor. In this paper, we construct a class of zero symmetric local near-ring of characteristic pk; k \(\ge\) 3 admitting an identity frobenius derivation, characterize the structures and orders of the set R(N), the regular compartment with an aim of advancing the classication problem of algebraic structures. The number theoretic notions relating the number of regular elements to Euler's phi-function and the arithmetic functions of Galois near-rings are adopted. Using the Fundamental Theorem of nitely generated Abelian groups, the structures of R(N) are proved to be isomorphic to cyclic groups of various orders. The study also extends to the automorphism groups Aut(R(N)) of the regular elements.

Порождение малых сложностных классов с помощью логических формул, схем и операций ограниченной конкатенации

The FO and FOM classes are small classes of recursive functions that characterize parallel computing with constant running time and a polynomial number of processors. The article provides an overview of the results describing the basic properties of the FO and FOM classes. Various ways of defining these classes and proving the equivalence of these definitions are given. The position of FO and FOM in the hierarchy of complexity classes is characterized. The results on the affiliation of various functions to the FO and FOM classes are presented. Superposition bases in the FO and FOM classes are described and a complete proof of the result on the superposition basis in the FOM class, consisting only of simple, mainly arithmetic functions, is given.

Resurgence, Stokes constants, and arithmetic functions in topological string theory

Resurgence, Stokes constants, and arithmetic functions in topological string theory

Polynomial Expressions for Certain Arithmetic Functions

t: We exhibit polynomial expressions for the arithmetic functions and , the number of representations of n as a sum of k squares and k triangular numbers, respectively, and also for the color partitions

Rounding error analysis of linear recurrences using generating series

We develop a toolbox for the error analysis of linear recurrences with constant or polynomial coefficients, based on generating series, Cauchy's method of majorants, and simple results from analytic combinatorics. We illustrate the power of the approach by several nontrivial application examples. Among these examples are a new worst-case analysis of an algorithm for computing Bernoulli numbers, and a new algorithm for evaluating differentially finite functions in interval arithmetic while avoiding interval blow-up.

Теоретико-числовi функцiї для цiлих ґаусових чисел

The classical number-theoretic functions – a number of divisors τ(n), sum of the divisors σ(n) and product of the divisors π(n) of a positive integer n – were generalized to the ring Z[i] of Gaussian integers. For the evaluation of the corresponding functions τ*(α), σ*m(α) and π*(α), obtained were the explicit formulae that use the canonical representation of α. A number of properties of these functions were studied, in particular, estimates from above for the functions τ*(α) and σ*m(α) and the properties connected with divisibility of their values by certain numbers. Researched are also sums of products of powers of the divisors for α∈Z[i].

Higher uniformity of arithmetic functions in short intervals I. All intervals

Abstract We study higher uniformity properties of the Möbius function $\mu $ , the von Mangoldt function $\Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$ . More precisely, letting $\Lambda ^\sharp $ and $d_k^\sharp $ be suitable approximants of $\Lambda $ and $d_k$ and $\mu ^\sharp = 0$ , we show for instance that, for any nilsequence $F(g(n)\Gamma )$ , we have $$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when $\theta = 5/8$ and $f \in \{\Lambda , \mu , d_k\}$ or $\theta = 1/3$ and $f = d_2$ . As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp \|_{U^s(X,X+H]}$ are also asymptotically small for any fixed s for these choices of $f,\theta $ . As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in $L^2$ . Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type $II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

The average behaviour of a hybrid arithmetic function associated to cusp form coefficients over certain sparse sequence

Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group ? = SL(2,Z), and let ?f (n), ?(n) and ?(n) be the nth normalized Fourier coefficient of the cusp form f , the sum-of-divisors function and the Euler totient function, respectively. In this paper, we investigate the asymptotic behaviour of the following summatory function Sj,b,c(x) := X n=a21 +a22 +a23 +a24 ?x (a1,a2,a3,a4)?Z4 ?j f (n)?b(n)?c(n), where j ? 2 is any given integer. In a similar manner, we also establish other similar results related to normalized coefficients of the symmetric power L-functions associated to holomorphic cusp form f.

ON ARITHMETIC FUNCTIONS, THEIR EXTENDED COEFFICIENTS: VARIOUS RESULTS AND RELATIONS

In this article, we represent a recurrence relation of the arithmetic function connected with an ascending factorial function, Lah and Stirling numbers. We then obtain a relation of harmonic numbers and again extend the coecients of these arithmetic functions involving Bell polynomials through introducing the sequence of Hankel type integrals. On the other hand, making some of the extensions of these arithmetic functions, we derive some more results and the summation formulae in terms of Riemann Zeta function.