A note on the Ramanujan τ-function

Abstract.

Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that

$$\# \{ \tau (n):n\, \leqslant\, x \} \gg x^{1/2} e^{ - 4\log x/ {\rm log} \log x}$$

We also show that

$$ \omega \left( {\mathop \prod \limits_{\begin{array}{*{20}c} {p \leqslant x} \\ {\tau (p) \ne 0} \\ \end{array} } \tau (p)\tau (p^2 )} \right) \gg \frac{{(\log x)^{13/11} }} {{\log \log x}}, $$

where ω(n) is the number of distinct prime divisors of n and p denotes prime numbers. These estimates improve several results from [6, 9].