Lehmer's conjecture for Ramanujan's tau function, $$ \Delta(q)=q\prod_{n=1}^\infty(1-q^n)^{24}=\sum_{m=1}^\infty\tau(m)q^m, $$ asserts that $\tau(m)$ never vanishes for $m=1,2,\dots$. In the recent question it was asked why it is important to have the nonvanishing.

I am wondering whether there are upper bounds, unconditional or conditional (modulo some other known conjectures), in terms of $x\in\mathbb R_+$ for the number of integers $m\le x$ satisfying $\tau(m)=0$ (maybe better, for the number of primes $p\le x$ satisfying $\tau(p)=0$)?

It looks like the series $\Delta(q)$ is very far from being "lacunary".
But besides Deligne's upper bound $|\tau(m)|\le d(m)m^{11/2}$
(where $d(\ )$ counts the number of divisors) and the lower bound
$$
\operatorname{card}\lbrace\tau(n):n\le x\rbrace\ge \operatorname{const}\cdot x^{1/2}e^{-4\log x/\log\log x}
$$
from
[M.Z. Garaev, V.C. Garcia, and S.V. Konyagin,
A note on the Ramanujan $\tau$-function, *Arch. Math. (Basel)* **89**:5 (2007) 411--418]
for the distribution of tau values, I cannot find any quantitative progress
towards Lehmer's original question.