Values of arithmetical functions equal to a sum of two squares

Abstract

Let wðnÞ denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n # x for which wðnÞ is the sum of two square numbers. We also obtain similar results for the Dedekind function cðnÞ and the sum of divisors function sðnÞ:

Compositions with the euler and carmichael functions

“…Now, given a large real number x, let y be defined implicitly by the equation x = exp 7s 1/2 log y , where s is defined as in (6). Then N T ≤ x holds for all such subsets T , provided that x is sufficiently large.…”

“…For a variety of other results with a similar flavor, we refer the reader to [3,4,5,6,8,10,12,13,21,22,23,24,26,33,34,40,42,45,49] and the references contained therein.…”

The least degree of a CM point on a modular curve

“…Most of our notation is standard, or will be introduced as necessary, but one exception is worth highlighting. For the remainder of the paper , we adopt the following convention for logarithms and iterated logarithms, borrowed from [3]. We write $$\ln {x}$$ for the usual natural logarithm and we set $$\begin{equation*} \log x := \max \lbrace 2,\ln x\rbrace .$$…”

Values of the Euler and Carmichael Functions which are Sums of Three Squares

“…The estimates (1) and (2) remain valid with (see [2, pp. 31, 43] and [4,Theorem 2]), and it is straightforward to prove that Theorem 1 also holds for . (See the remarks following the proof of the Theorem 3, which is a generalization of Theorem 1.)…”

“…(See the remarks following the proof of the Theorem 3, which is a generalization of Theorem 1.) One can also show that (1) and (2) hold with ' replaced by (see [2,Theorem 6.3 and §7] and [3]). For sums of three squares, we can prove the following:…”