O número médio de representações de um inteiro positivo como soma dos quadrados de dois inteiros

Abstract

This paper aims to address two themes: the representation of positive integers as sum of squares and the average number of representations of a positive integer as the sum of two squares. About the first theme, we will prove several results to understand under what conditions a positive integer has a representation as a sum of two, three or four squares. About the second theme, we will prove that the mean number of representations of a positive integer as the sum of the squares of two integers is . To do so, we will introduce the function s 2 which associates an integer n with the cardinality of the set X n = {( a, b ) ∈ Z 2 ; a 2 + b 2 = n } and we will calculate the limit of its average value. Finally, as an analogy to the result regarding the mean value of s 2 , we will define the function s 3 , that associates a positive integer n with the cardinality of the set Y n = {( a, b, c ) ∈ Z 3 ; a 2 + b 2 + c 2 = n } and we will prove that there is no mean number of representations of a positive integer as the sum of the squares of three integers.