Asymptotic behavior of the irrational factor

Abstract

We study the irrational factor function I(n) introduced by Atanassov and dened by I(n)is the prime factorization of n. We show that the sequence G(n)/n n 1 , where G(n) = n ν=1 I(ν) 1/n , is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).

A CLASS OF ARITHMETIC FUNCTIONS ON PSL2(Z)

“…In [1], Alkan, Ledoan and one of the authors describe a precise asymptotic for the function G(n), and establish further results showing that the function I(n) is very regular on average.…”

A CLASS OF ARITHMETIC FUNCTIONS ON PSL2(ℤ), II

“…In the present paper, we continue the work done in [11], namely, a study of a class of arithmetic functions that generalize the so-called irrational factor function I(n) and the strong restrictive factor function R(n), which are defined in [2] and [3] by I(n) = Panaitopol, Alkan, Ledoan, and the authors develop a number of results on these arithmetic functions ( [1], [8], [9], and [10]). In the prequel, the authors establish results on the average values of the functions f A (n) = p α ||n p aα+b cα+d for a class of matrices A = a b c d in PSL 2 (Z).…”

“…We remark that if λ is such that the above maximum is attained at both x 0 and x 0 , where x 0 is not an integer, then F A,λ (s) has a double pole at s = θ (1) . Furthermore, we note that for a given matrix A, the set of such exceptional λ is at most countable.…”

“…the main contribution being due to the residue of the simple pole at the point s = θ (1) . In order to estimate the integral along the modified contour one makes use of the bounds…”

“…We remark that given a matrix A there may possibly be a finite or countable set of λ for which F A,λ (s) has a double pole at s = θ (1) . In these rare cases M A,λ (x) has a different order of magnitude.…”