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

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].

On the Davison Convolution of Arithmetical Functions

1989 ◽  
Vol 32 (4) ◽  
pp. 467-473 ◽  
Author(s):  
Pentti Haukkanen
Keyword(s):  
Positive Integer ◽  
Arithmetical Functions ◽  
Positive Divisor ◽  
Complex Valued ◽  
Ordered Pairs

AbstractThe Davison convolution of arithmetical functions f and g is defined by where K is a complex-valued function on the set of all ordered pairs (n, d) such that n is a positive integer and d is a positive divisor of n. In this paper we shall consider the arithmetical equations f(r) = g, f(r) = fg, f o g = h in f and the congruence (f o g)(n) = 0 (mod n), where f(r) is the iterate of f with respect to the Davison convolution.

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

1929 ◽  
Vol 25 (3) ◽  
pp. 255-264 ◽  
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.

scholarly journals Arithmetical functions commutable with sums of squares

2021 ◽  
Vol 27 (3) ◽  
pp. 143-154
Author(s):  
I. Kátai ◽  
◽  
B. M. Phong ◽  
Keyword(s):  
Sums Of Squares ◽  
Complex Numbers ◽  
Arithmetical Functions ◽  
Positive Integers

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.

GENERALIZED ARITHMETICAL FUNCTIONS OF THREE VARIABLES

2010 ◽  
Vol 06 (07) ◽  
pp. 1689-1699 ◽  
Author(s):  
EMIL DANIEL SCHWAB
Keyword(s):  
Arithmetical Functions ◽  
Incidence Algebra ◽  
Positive Integers

The paper is devoted to the study of some properties of generalized arithmetical functions extended to the case of three variables. The convolution in this case is a convolution of the incidence algebra of a Möbius category in the sense of Leroux. This category is a two-sided analogue of the poset (it is viewed as a category) of positive integers ordered by divisibility.

scholarly journals I–Convergence of Arithmetical Functions

2020 ◽  
Author(s):  
Vladimír Baláž ◽  
Tomáš Visnyai
Keyword(s):  
Statistical Convergence ◽  
Canonical Representation ◽  
Asymptotic Density ◽  
Normal Order ◽  
Arithmetical Functions ◽  
Density Zero ◽  
Positive Integers ◽  
The Ideal

Let n > 1 be an integer with its canonical representation, n = p 1 α 1 p 2 α 2 ⋯ p k α k . Put H n = max α 1 … α k , h n = min α 1 … α k , ω n = k , Ω n = α 1 + ⋯ + α k , f n = ∏ d ∣ n d and f ∗ n = f n n . Many authors deal with the statistical convergence of these arithmetical functions. For instance, the notion of normal order is defined by means of statistical convergence. The statistical convergence is equivalent with I d –convergence, where I d is the ideal of all subsets of positive integers having the asymptotic density zero. In this part, we will study I –convergence of the well-known arithmetical functions, where I = I c q = A ⊂ N : ∑ a ∈ A a − q < + ∞ is an admissible ideal on N such that for q ∈ 0 1 we have I c q ⊊ I d , thus I c q –convergence is stronger than the statistical convergence ( I d –convergence).

scholarly journals Set of uniqueness of shifted Gaussian primes

2015 ◽  
Vol 53 (1) ◽  
pp. 123-133
Author(s):  
Jay Mehta ◽  
G.K. Viswanadham
Keyword(s):  
Set Of Uniqueness ◽  
Gaussian Primes

scholarly journals On the rational solutions of y^2 =x^3 + k^{6n+3}

2021 ◽  
Vol 27 (3) ◽  
pp. 130-142
Author(s):  
Richa Sharma ◽  
◽  
Sanjay Bhatter ◽  
Keyword(s):  
Complex Numbers ◽  
Rational Solutions ◽  
Arithmetical Functions ◽  
Positive Integers

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.

A UNIQUE FACTORIZATION IN COMMUTATIVE MÖBIUS MONOIDS

2008 ◽  
Vol 04 (04) ◽  
pp. 549-561 ◽  
Author(s):  
EMIL DANIEL SCHWAB ◽  
PENTTI HAUKKANEN
Keyword(s):  
Fundamental Theorem ◽  
Factorization Theorem ◽  
Unique Factorization ◽  
Multiplicative Semigroup ◽  
Arithmetical Functions ◽  
Bicyclic Semigroup ◽  
Fundamental Theorem Of Arithmetic ◽  
Positive Integers ◽  
Special Case

We show that any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Particular attention is paid to standard examples, which arise from the bicyclic semigroup and the multiplicative analogue of the bicyclic semigroup. The second example shows that the Fundamental Theorem of Arithmetic is a special case of the unique factorization theorem in commutative Möbius monoids. As an application, we study generalized arithmetical functions defined on an arbitrary commutative Möbius monoid.

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