INDUCTIVE FORMULAS FOR SOME ARITHMETIC FUNCTIONS

Some Asymptotic Formulas for Multiplicative Functions

Some Asymptotic Formulas on Generalized Divisor Functions

1 . Throughout this paper, we use the following notation : c•1 , c2 , . . ., X0 , X1 , . . . denote positive absolute constants. We denote the number of elements of the finite set S by BSI . We write ex =exp (x) . We denote the least prime factor of n by p(n) . We write pall n if pain but pa+1 f n . v(n) denotes the number of the distinct prime factors of n, while the number of all the prime fa...

On the Iterates of Some Arithmetic Functions

1. Introduction. For any arithmetic function f(n), we denote its iterates as follows : f l (n) = f (n) , fk (n) = fl[ fk_1(n) ] (k > 1). Let a(n) and a* (n) denote,'respectívely, the sum of the divisors of n, and the sum of its unitary divisors, where we recall that d is called a unitary divisor of n if (d,n/d) = 1. Makowski and Schinzel [3] proved that and conjectured that ak (n) lim inf n < o...

On the Coprimality of Some Arithmetic Functions

Let φ stand for the Euler function. Given a positive integer n, let σ(n) stand for the sum of the positive divisors of n and let τ(n) be the number of divisors of n. We obtain an asymptotic estimate for the counting function of the set {n : gcd(φ(n), τ(n)) = gcd(σ(n), τ(n)) = 1}. Moreover, setting l(n) := gcd(τ(n), τ(n+ 1)), we provide an asymptotic estimate for the size of #{n 6 x : l(n) = 1}.

Braids, Galois Groups, and Some Arithmetic Functions

This lecture is about some new relations among the classical objects of the title. The study of such relations was started by [Bi, G, De, Ihj] from independent motivations, and was developed in [A3, C3, A-I, IKY, Dr2, O, N], etc. It is still a very young subject, and there are several different approaches, each partly blocked by its own fundamental conjectures! But it is already allowing one to...