On Some Properties of the Hofstadter-Mertens Function

Abstract

Many mathematicians have been interested in the study of recursive sequences. Among them, a class of "chaotic" sequences are named "meta-Fibonacci sequences." The main example of meta-Fibonacci sequence was introduced by Hofstadter, and it is called the Q-sequence. Recently, Alkan-Fox-Aybar and the author studied the pattern induced by the connection between the Q-sequence and other known sequences. Here, we continue this program by studying a "Mertens' version" of the Hofstadter sequence, defined (for x > 0) by x ->Sigma(n <= x)mu(n)Q(n), where mu(n) is the Mobius function. In particular, as we shall see, this function encodes many interesting properties which relate prime numbers to "meta-sequences".