The Sum of Reciprocals of Divisors of Lucas Sequences

Abstract

In this thesis we extend results on reciprocal sums of divisors of Lucas sequences due to Pollack, Engberg and Agrawal to results that are similar in spirit for reciprocal sums of divisors of companion Lucas sequences. Assuming the generalised Riemann Hypothesis and the we Elliot-Halberstam conjecture, we find the maximal order for the reciprocal sum of all prime divisors of a companion Lucas sequence and then move finding the maximal order for the reciprocal sum of all divisors of a companion Lucas sequence. We then follow this up by proving an asymptotic for the sum of reciprocals of all primitive divisors of a companion Lucas sequence while assuming only the generalised Riemann Hypothesis. After this we prove, unconditionally, that the function mapping an integer n to the reciprocal sum of the (prime) divisors of the n-th entry in a companion Lucas sequence has a continuous distribution function. In order to prove these statements we first discuss some required theory on smooth numbers, probabilistic number theory and Lucas sequences. After which we apply this knowledge to prove a version of Mertens' Theorem that is suitable for our purposes. We then finish by proving the aforementioned theorems.