scholarly journals Arithmetical Functions Commutable with Sums of Squares II

2021 ◽  
Author(s):  
Imre Kátai ◽  
Bui Minh Phong
Keyword(s):  
Complex Number ◽  
Sums Of Squares ◽  
Multiplicative Functions ◽  
Arithmetical Functions

We give all functions ƒ , E: ℕ → ℂ which satisfy the relation for every a, b, c ∈ ℕ, where h ≥ 0 is an integers and K is a complex number. If n cannot be written as a2 + b2 + c2 + h for suitable a, b, c ∈ ℕ, then ƒ (n) is not determined. This is more complicated if we assume that ƒ and E are multiplicative functions.

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.

scholarly journals Multiplicative functions commutable with sums of squares

2018 ◽  
Vol 14 (02) ◽  
pp. 469-478 ◽  
Author(s):  
Poo-Sung Park
Keyword(s):  
Multiplicative Function ◽  
Sums Of Squares ◽  
Multiplicative Functions ◽  
Identity Function ◽  
Positive Integers

Let [Formula: see text] be an integer greater than or equal to [Formula: see text]. We show that if a multiplicative function [Formula: see text] satisfies [Formula: see text] for all positive integers [Formula: see text], then [Formula: see text] is the identity function.

scholarly journals A lower bound for the variance of generalized divisor functions in arithmetic progressions

2021 ◽  
Author(s):  
Daniele Mastrostefano
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Complex Number ◽  
Arithmetic Progressions ◽  
Divisor Function ◽  
Multiplicative Functions ◽  
Divisor Functions

AbstractWe prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $$\alpha $$ α -fold divisor function, for any complex number $$\alpha \not \in \{1\}\cup -\mathbb {N}$$ α ∉ { 1 } ∪ - N , even when considering a sequence of parameters $$\alpha $$ α close in a proper way to 1. Our work builds on that of Harper and Soundararajan, who handled the particular case of k-fold divisor functions $$d_k(n)$$ d k ( n ) , with $$k\in \mathbb {N}_{\ge 2}$$ k ∈ N ≥ 2 .

scholarly journals On the coefficients in the expansions of certain modular functions

1918 ◽  
Vol 95 (667) ◽  
pp. 144-155 ◽  
Keyword(s):  
Unit Circle ◽  
New Method ◽  
Sums Of Squares ◽  
Additional Property ◽  
Modular Functions ◽  
Natural Boundary ◽  
Arithmetical Functions ◽  
Order Of Magnitude ◽  
Restricted Region

1. A very large proportion of the most interesting arithmetical functions —of the functions, for example, which occur in the theory of partitions, the theory of the divisors of numbers, or the theory of the representation of numbers by sums of squares—occur as the coefficients in the expansions of elliptic modular functions in powers of the variable q = e π i τ . All of these functions have a restricted region of. existence, the unit circle | q | = 1 being a “ natural boundary” or line of essential singularities. The most important of them, such as the functions (ω 1 /π) 12 ∆ = q 2 {(1- q 2 ) (1- q 4 )...} 24 , (1, 1) ϑ 3 (0) = 1 + 2 q + 2 q 4 + 2 q 9 + ....., (1. 2) 12 (ω 1 /π) 4 g 2 = 1 + 240 (1 3 q 2 /1- q 2 + 2 3 q 4 /1- q 4 + ...), (1, 3) 216 (ω 1 /π) 6 g 3 = 1 - 504 (1 5 q 2 /1- q 2 + 2 5 q 4 /1- q 4 + ...), (1, 4) are regular inside the unit circle ; and many, such as the functions (1, 1) and (1, 2), have the additional property of having no zeros inside the circle, so that their reciprocals are also regular. In a series of recent papers we have applied a new method to the study of these arithmetical functions. Our aim has been to express them as series which exhibit explicitly their order of magnitude, and the genesis of their irregular variations as n increases. We find, for example, for p ( n ) the number of unrestricted partitions of n ,and for r s ( n ), the number of repre­sentations of n as the sum of an even number s of squares, the series

Square-Reduced Residue Systems (Mod r) and Related Arithmetical Functions

1979 ◽  
Vol 22 (2) ◽  
pp. 207-220 ◽  
Author(s):  
R. Sivaramakrishnan
Keyword(s):  
Symmetric Functions ◽  
Arithmetic Function ◽  
Exponential Sum ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Class Division ◽  
Arithmetical Functions ◽  
New Class ◽  
Image Position ◽  
Residue Systems

AbstractWe define a square-reduced residue system (mod r) as the set of integers a (mod r) such that the greatest common divisor of a and r, denoted by (a, r), is a perfect square ≥ 1 and contained in a residue system (mod r). This leads to a Class-division of integers (mod r) based on the 'square-free' divisors of r. The number of elements in a square-reduced residue system (mod r) is denoted by b(r). It is shown that(1)(2)In view of (2), b(r) is said to be 'specially multiplicative'. The exponential sum associated with a square-reduced residue system (mod r) is defined bywhere the summation is over a square-reduced residue system (mod r).B(n, r) belongs to a new class of multiplicative functions known as 'Quasi-symmetric functions' and(3)As an application, the sum is considered in terms of the Cauchy-composition of even functions (mod r). It is found to be multiplicative in r. The evaluation of the above sum gives an identity involving Pillai's arithmetic function

Arithmetical Functions

1994 ◽  
Author(s):  
Wolfgang Schwarz ◽  
Jürgen Spilker
Keyword(s):  
Arithmetical Functions

scholarly journals Towards a computational proof of Vizing's conjecture using semidefinite programming and sums-of-squares

2021 ◽  
Vol 107 ◽  
pp. 67-105
Author(s):  
Elisabeth Gaar ◽  
Daniel Krenn ◽  
Susan Margulies ◽  
Angelika Wiegele
Keyword(s):  
Semidefinite Programming ◽  
Sums Of Squares
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