TITLE: Lehmer's conjecture for Ramanujan's tau function
QUESTION [12 upvotes]: 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.

REPLY [3 votes]: Lehmer's conjecture has an equivalent result in the theory of Harmonic Maass forms.  The non-vanishing of the tau function is equivalent to the irrationality of the coefficients of Harmonic Maass forms.  
Specifically there is a correspondence between the spaces $ \zeta_{2-k} : H_{2-k}(N, \chi) \rightarrow S_k(N, \chi) $. where 

$ \zeta_{2-k}$ is a differential operator
H = Harmonic Maass forms
S = cusp forms (referred to as the shadow of the Maass form)

The discriminant function $\Delta(z)$ is the shadow of the Harmonic Maass form $\frac{1}{11!} Q^+(-1, 12, 1; z) $  
See Theorem 12.5 in the paper Unearthing the visions of a master: harmonic Maass forms and number theory by Ken Ono.  Also see Algebraicity of Harmonic Maass forms