We prove a number of results regarding odd values the Ramanujan $$\tau $$ -function. For example, we existence an effectively computable positive constant $$\kappa such that if (n)$$ is and $$n \ge 25$$ then either $$\begin{aligned} P(\tau (n)) \; > \kappa \cdot \frac{\log \log {n}}{\log {n}} \end{aligned}$$ or there exists prime $$p \mid n$$ with (p)=0$$ . Here P(m) denotes largest factor m. a...
