Arithmetic properties of the Ramanujan function

Abstract

We study some arithmetic properties of the Ramanujan function $\tau (n)$, such as the largest prime divisor $P(\tau (n))$ and the number of distinct prime divisors $\omega (\tau (n))$ of $\tau (n)$ for various sequences of $n$. In particular, we show that \hbox{$P(\tau (n))\ge (\log n{)}^{33/31+o(1)}$} for infinitely many $n$, and \begin{equation*} P(τ(p)τ(p^2)τ(p^3)) > (1+o(1))\frac{\log\log p\log\log\log p} {\log\log\log\log p} \end{equation*} for every prime $p$ with \hbox{$\tau (p)\ne 0$}.

8 pages