Arithmetical functions commutable with sums of squares

Let k\in{\mathbb N}_0 and K\in \mathbb C, where {\mathbb N}_0, \mathbb C denote the set of nonnegative integers and complex numbers, respectively. We give all functions f, h_1, h_2, h_3, h_4:{\mathbb N}_0\to \mathbb C which satisfy the relation \[f(x_1^2+x_2^2+x_3^2+x_4^2+k)=h_1(x_1)+h_2(x_2)+h_3(x_3)+h_4(x_4)+K\] for every x_1, x_2, x_3, x_4\in{\mathbb N}_0. We also give all arithmetical functions F, H_1, H_2, H_3, H_4:{\mathbb N}\to \mathbb C which satisfy the relation \[F(x_1^2+x_2^2+x_3^2+x_4^2+k)=H_1(x_1)+H_2(x_2)+H_3(x_3)+H_4(x_4)+K\] for every x_1,x_2, x_3,x_4\in{\mathbb N}, where {\mathbb N} denotes the set of all positive integers.

Download Full-text

Empirical Theorems in Diophantine Analysis

Mathematics Teacher ◽

10.5951/mt.16.5.0257 ◽

1923 ◽

Vol 16 (5) ◽

pp. 257-265

Author(s):

R. D. Carmichael

Keyword(s):

The State ◽

Sums Of Squares ◽

Carnegie Institution ◽

Comprehensive Account ◽

History Of ◽

Positive Integers ◽

Theory Of Numbers

The larger portion of the theorems in Diophantine Analysis probably existed first as empirical or conjectural theorems. Many of them passed to the state of proved theorems before they left the hands of those who discovered them; many others were proved in the same generation in which they were made public; not a few required a longer period for their proof; and several remain today as a silent challenge to the skill and power of contemporary mathematicians. The remarks may be illustrated with a brief account of the history of the problem of representing numbers (that is, positive integers) as sums of squares of integers and of higher powers. Anyone interested in further details will find them in the comprehensive account of Diophantine Analysis which fills volume II (xxvi + 803 pages) of L. E. Dickson's “History of the Theory of Numbers,” Carnegie Institution, Washington, D. C. We shall make free use of the material summarized in a masterly way in this volume.

Download Full-text

Quasi-uniqueness of the set of "Gaussian prime plus one's"

International Journal of Number Theory ◽

10.1142/s1793042114500559 ◽

2014 ◽

Vol 10 (07) ◽

pp. 1783-1790

Author(s):

Jay Mehta ◽

G. K. Viswanadham

Keyword(s):

Acta Arith ◽

Gaussian Integers ◽

Arithmetical Functions ◽

Set Of Uniqueness ◽

Positive Integers ◽

Complex Valued ◽

Gaussian Primes

We recall the well-known notion of the set of uniqueness for arithmetical functions, introduced by Kátai and several other mathematicians like Indlekofer, Elliot and Hoffman, independently. We define its analogue for completely additive complex-valued functions over the set of non-zero Gaussian integers with some examples. We show that the set of "Gaussian prime plus one's" along with finitely many Gaussian primes of norm up to some constant K is a set of uniqueness with respect to Gaussian integers. This is analogous to Kátai's result in the case of positive integers [I. Kátai, On sets characterizing number theoretical functions, II, Acta Arith.16 (1968) 1–14].

Download Full-text

On Lacunary Mean Ideal Convergence in Generalized Randomn-Normed Spaces

Abstract and Applied Analysis ◽

10.1155/2014/101782 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 2

Author(s):

Awad A. Bakery ◽

Mustafa M. Mohammed

Keyword(s):

Normed Space ◽

Complex Numbers ◽

Normed Spaces ◽

Ideal Convergence ◽

Cauchy Sequences ◽

Limit Points ◽

Positive Integers ◽

Cluster Points

An idealIis a hereditary and additive family of subsets of positive integersℕ. In this paper, we will introduce the concept of generalized randomn-normed space as an extension of randomn-normed space. Also, we study the concept of lacunary mean (L)-ideal convergence andL-ideal Cauchy for sequences of complex numbers in the generalized randomn-norm. We introduceIL-limit points andIL-cluster points. Furthermore, Cauchy andIL-Cauchy sequences in this construction are given. Finally, we find relations among these concepts.

Download Full-text

SUMS OF SQUARES AND PARTITION CONGRUENCES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788720000117 ◽

2020 ◽

pp. 1-19

Author(s):

SU-PING CUI ◽

NANCY S. S. GU

Keyword(s):

Positive Integer ◽

Sums Of Squares ◽

Binomial Coefficients ◽

Partition Congruences ◽

Arithmetic Properties ◽

Positive Integers ◽

Pair Function

For positive integers $n$ and $k$ , let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$ , by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$ . Furthermore, in view of these arithmetic properties of $r_{k}(n)$ , we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

Download Full-text

On Ramanujan's Arithmetical Function Σr, s(n)

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100013943 ◽

1929 ◽

Vol 25 (3) ◽

pp. 255-264 ◽

Cited By ~ 5

Author(s):

J. R. Wilton

Keyword(s):

Error Term ◽

Arithmetical Function ◽

Arithmetical Functions ◽

Image Position ◽

The Right ◽

Positive Integers

Let σs(n) denote the sum of the sth powers of the divisors of n,and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, ifandthen , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the formfor every ε > 0, and that it is not of the form . He further conjectures thatfor all positive values of r and s; and this conjecture has recently been proved to be correct.

Download Full-text

Exponentials and Logarithms Properties in an Extended Complex Number Field

10.20944/preprints202104.0207.v4 ◽

2021 ◽

Author(s):

Daniel Tischhauser

Keyword(s):

Complex Number ◽

Number System ◽

Complex Numbers ◽

Binary Operations ◽

Algebraic Properties ◽

Extended Complex ◽

Multivalued Functions ◽

Positive Integers ◽

Complex Exponential ◽

Complex Number Field

It is well established the complex exponential and logarithm are multivalued functions, both failing to maintain most identities originally valid over the positive integers domain. Moreover the general case of complex logarithm, with a complex base, is hardly mentionned in mathematic litterature. We study the exponentiation and logarithm as binary operations where all operands are complex. In a redefined complex number system using an extension of the C field, hereafter named E, we prove both operations always produce single value results and maintain the validity of identities such as logu (w v) = logu (w) + logu (v) where u, v, w in E. There is a cost as some algebraic properties of the addition and subtraction will be diminished, though remaining valid to a certain extent. In order to handle formulas in a C and E dual number system, we introduce the notion of set precision and set truncation. We show complex numbers as defined in C are insufficiently precise to grasp all subtleties of some complex operations, as a result multivaluation, identity failures and, in specific cases, wrong results are obtained when computing exclusively in C. A geometric representation of the new complex number system is proposed, in which the complex plane appears as an orthogonal projection, and where the complex logarithm an exponentiation can be simply represented. Finally we attempt an algebraic formalization of E.

Download Full-text

Development of mandelbrot set for the logistic map with two parameters in the complex plane

Eastern-European Journal of Enterprise Technologies ◽

10.15587/1729-4061.2021.249264 ◽

2021 ◽

Vol 6 (3 (114)) ◽

pp. 47-56

Author(s):

Wasan Saad Ahmed ◽

Saad Qasim Abbas ◽

Muntadher Khamees ◽

Mustafa Musa Jaber

Keyword(s):

Fixed Points ◽

Critical Point ◽

Complex Plane ◽

Dynamical Behavior ◽

Logistic Map ◽

Mandelbrot Set ◽

Complex Numbers ◽

Iteration Function ◽

Two Parameters ◽

Positive Integers

In this paper, the study of the dynamical behavior of logistic map has been disused with representing fractals graphics of map, the logistic map depends on two parameters and works in the complex plane, the map defined by f(z,α,β)=αz(1–z)β. where and are complex numbers, and β is a positive integers number, the visualization method used in this work to generate fractals of the map and to inspect the relation between the value of β and the shape of the map, this visualization analysis showed also that, as the value of β increasing, as the number of humps in the function also increasing, and it demonstrate that is true also for the function’s first iteration , f2(x0)=f(f(x0)) and the second iteration , f3(x0)=f(f2(x0)), beside that , the visualization technique showed that the number of humps in that fractal is less than the ones in the second iteration of the original function ,the study of the critical points and their properties of the logistic map also discussed it, whereas finding the fixed point led to find the critical point of the function f, in addition , it haven proven for the set of all pointsα∈C and β∈N, the iteration function f(f(z) has an attractive fixed points, and belongs to the region specified by the disc |1–β(α–1)|<1. Also, The discussion of the Mandelbrot set of the function defined by the f(f(z)) examined in complex plans using the path principle, such that the path of the critical point z=z0 is restricted, finally, it has proven that the Mandelbrot set f(z,α,β) contains all the attractive fixed points and all the complex numbers in which α≤(1/β+1) (1/β+1) and the region containing the attractive fixed points for f2(z,α,β) was identified

Download Full-text