Arithmetic functions with linear recurrence sequences

Abstract

We obtain asymptotic formulas for all the moments of certain arithmetic functions with linear recurrence sequences. We also apply our results to obtain asymptotic formulas for some mean values related to average orders of elements in finite fields.

Multiperfect numbers on lines of the Pascal triangle

“…When (u n ) n 0 is a nondegenerate linearly recurrent sequence of integers, then the average value of σ (|u n |)/|u n | when n ranges in the interval [1, x] through values such that u n = 0 approaches a finite limit when x → ∞. This is a result of Shparlinski from [22] (see also [17]). For binomial coefficients, the situation is different.…”

The reciprocal sum of divisors of Mersenne numbers

“…We expect U n -analogues of all of our theorems. For Theorem 1.4, these analogues are fairly simple to show (and the case P = 1, Q = −1, corresponding to the Fibonacci numbers, was already discussed): Again [18] gives existence of the distribution function. It is known [2] that for each n > 30 there is a prime p with z(p) = n. This allows us to deduce continuity from the variant of Theorem 5.4 discussed above.…”

“…In [8, p. 270], Erdős, Kiss, and Pomerance claim that it can be proved by "more or less standard methods" that f (n) possesses a continuous distribution function. That f (n) and F (n) have distribution functions follows immediately from (later) general results of Luca and Shparlinski [18], who used the method of moments. (See [17, proof of Theorem 3(2)] for an alternative approach to these results.)…”

“…(a) Let a(n) = p|Fn 1/p, where F n is the nth Fibonacci number. It follows from the general results of [18] that a(n) has a distribution function. To show continuity, we apply the variant of Theorem 5.4 just discussed.…”

On Quotients of Values of Euler's Function on Factorials

“…), where d is the divisor counting function and σ is summatory functions for divisors. In [6], Luca and Shparlinski obtained asymptotic formulas for moments of certain arithmetic functions with linear recurrence sequences. Further, Luca in [5] considered ϕ(F n ), where F n is the nth Fibonacci number.…”