Divisor Sums of Generalised Exponential Polynomials

Abstract

A study is made of sums of reciprocal norms of integral and prime ideal divisors of algebraic integer values of a generalised exponential polynomial. This includes the important special cases of linear recurrence sequences and general sums of S-units. In the case of an integral binary recurrence sequence, similar (but stronger) results were obtained by P. Erdős, P. Kiss and C. Pomerance.

Counting the values taken by algebraic exponential polynomials

“…Considerable interest has been shown in the arithmetic of these exponential polynomials in the case when r = 1, see [3], [7], [8], [9], [11], [12], [13]. Often, quite simply stated questions have turned out to be particularly intractable by elementary methods.…”

Arithmetic functions with linear recurrence sequences

“…It has also been shown in [3,15] that for ν = 1 and if the function ξ(k) is supported only on prime numbers (as in the case of the functions f (k), g(k), h(k) above), the error term in the above mentioned asymptotic formulas for S 1 (ϑ, M, N) can be estimated substantially better than in the general case. However, this approach does not immediately extend to ν 2.…”

“…However, for ν = 1 and a very slowly growing function ξ , asymptotic formulas for S 1 (ϑ, M, N) have been obtained in [3,15]. For instance, such functions as…”

“…Here, we use the same approach as in [3,15] to obtain asymptotic formulas for S ν (ϑ, M, N) given by (1) and for any fixed ν 1. It has also been shown in [3,15] that for ν = 1 and if the function ξ(k) is supported only on prime numbers (as in the case of the functions f (k), g(k), h(k) above), the error term in the above mentioned asymptotic formulas for S 1 (ϑ, M, N) can be estimated substantially better than in the general case.…”