On the prime factors of the iterates of the Ramanujan $\tau$--function

TL;DR

This paper investigates the size and prime factors of the iterates of the Ramanujan tau function for positive integers, aiming to understand their arithmetic properties and growth behavior.

Contribution

It provides new insights into the prime factorization patterns and size estimates of the iterated tau function values, a topic less explored in previous research.

Findings

Identification of prime factorization patterns in tau iterates

Bounds on the size of tau iterates for large n

Results on the distribution of prime factors in tau iterates

Abstract

In this paper, for a positive integer $n\ge 1$, we look at the size and prime factors of the iterates of the Ramanujan $\tau$ function applied to $n$.