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.
Best Answer
One of the canonical references for questions like this is Serre's "Quelques applications du theoreme de densite de Chebotarev", Publ. Math. IHES 54. He proves, for example, that the number of primes $0\leq p \leq X$ with $\tau(p)=0$ is $\ll X (\log{X})^{-3/2}$ unconditionally, and is $\ll X^{\frac{3}{4}}$ under GRH.