On arithmetic functions and divisors of higher order

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...

On the Normal Order of Arithmetic Functions.

Concerning Some Arithmetic Functions Which Use Exponential Divisors

Let σ(e)(n) denote the sum of the exponential divisors of n, τ (e)(n) denote the number of the exponential divisors of n, σ(e)∗(n) denote the sum of the e-unitary divisors of n and τ (e)∗(n) denote the number of the e-unitary divisors of n. The aim of this paper is to present several inequalities about the arithmetic functions which use exponential divisors. Among these inequalities, we have th...

Hodge Index Theorem for Arithmetic Divisors with Degenerate Green Functions

As we indicated in our paper [9], the standard arithmetic Chow groups introduced by Gillet-Soulé [3] are rather restricted to consider arithmetic analogues of geometric problems. In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem still holds as in papers [1], [7] and [14]. Let X → Spec(Z) be a regular ari...

Higher order cohomology of arithmetic groups

Higher order cohomology of arithmetic groups is expressed in terms of (g, K)-cohomology. It is shown that the latter can be computed using functions of moderate growth. Higher order versions of results of Borel are proven and the Borel conjecture in the higher order setting is stated.