The Average Number of Divisors in Certain Arithmetic Sequences

On certain arithmetic functions involving exponential divisors

The integer d is called an exponential divisor of n = ∏r i=1 p ai i > 1 if d = ∏r i=1 p ci i , where ci|ai for every 1 ≤ i ≤ r. The integers n = ∏r i=1 p ai i ,m = ∏r i=1 p bi i > 1 having the same prime factors are called exponentially coprime if (ai, bi) = 1 for every 1 ≤ i ≤ r. In this paper we investigate asymptotic properties of certain arithmetic functions involving exponential divisors a...

The Average Number of Divisors of the Euler Function

The upper bound and the lower bound of average numbers of divisors of Euler Phi function and Carmichael Lambda function are obtained by Luca and Pomerance (see [LP]). We improve the lower bound and provide a heuristic argument which suggests that the upper bound given by [LP] is indeed close to the truth.

On the average number of divisors of the Euler function

We let φ(·) and τ(·) denote the Euler function and the number-ofdivisors function, respectively. In this paper, we study the average value of τ(φ(n)) when n ranges in the interval [1, x].

On the average number of sharp crossings of certain Gaussian random polynomials

A remark on the means of the number of divisors

‎We obtain the asymptotic expansion of the sequence with general term $frac{A_n}{G_n}$‎, ‎where $A_n$ and $G_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$‎, ‎with $d(n)$ denoting the number of positive divisors of $n$‎. ‎Also‎, ‎we obtain some explicit bounds concerning $G_n$ and $frac{A_n}{G_n}$.