On Some Properties of the Hofstadter–Mertens Function

Pavel Trojovský

doi:10.1155/2020/1816756

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↦∑n≤xμnQn, where µ(n) is the Möbius function. In particular, as we shall see, this function encodes many interesting properties which relate prime numbers to “meta-sequences”.

Highlights

At first glance, the “self-definition” of Q(n) appears to be a very strange definition
We continue the fruitful program started by Hofstadter, Golomb, Pinn, Alkan, Fox, and Aybar to study the behavior of meta-Fibonacci sequences, mainly the Q-sequence
MQ(n) which is defined as the accumulation function of the product between the Q-sequence and the Mobius function (we call MQ(n) as Hofstadter–Mertens function). e sequence MQ(n) is studied with emphasis on its chaotic behavior

Summary

Pavel Trojovsky

The first class of recursive sequences without a fixed order, was proposed, in 1979, by Hofstadter and Godel [1] They defined (Q(n))n≥1 by the self-recurrence relation as follows: Q(n) Q(n − Q(n − 1)) + Q(n − Q(n − 2)), (1). The structure of such a function is strongly reflected in its growth properties For this reason, this section will be devoted to this kind of study. E growth behavior of the graphical structure of MQ(n) brings a complex fractal-like structure These kinds of patterns are commonly called “generational structure” of a meta-Fibonacci sequence (for more information about these structures for other metaFibonacci sequences, we refer the reader to [9,10,11]).

Standard deviation

Conclusion