Random multiplicative walks on the residues modulo n

TL;DR

This paper introduces a new arithmetic function related to random multiplicative processes modulo n, analyzes its asymptotic behavior, and explores its relationship with the largest prime divisor and the second largest divisor of n.

Contribution

It defines and studies the properties of a novel function a(n), revealing its asymptotic relation to P_1(n) and P_2(n), and provides formulas and asymptotic analysis.

Findings

a(n) and P_1(n) have the same average order asymptotically

the difference a(n)-P_1(n) tends to infinity outside a set of density zero

the sum of differences relates asymptotically to the sum of second largest divisors

Abstract

We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to 0 modulo $n$. We give several formulas for computing the values of this function and analyze its asymptotic behavior. We find that it is closely related to $P_1(n)$, the largest prime divisor of $n$. In particular, $a(n)$ and $P_1(n)$ have the same average order asymptotically. Furthermore, the difference between the functions $a(n)$ and $P_1(n)$ is $o(1)$ as $n$ tends to infinity on a set with density approximately $0.623$. On the other hand however, we see that (except on a set of density zero) the difference between $a(n)$ and $P_1(n)$ tends to infinity on the integers outside this set. Finally we consider the asymptotic behaviour of the difference between these two functions and find that $\sum_{n\leq x}\big( a(n)-P_1(n)\big) \sim \left(1-\frac{\pi}{4}\right)\sum_{n\leq x} P_2(n)$, where $P_2(n)$ is the second largest divisor of $n$.