Question

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.

Answer 1

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.

Answer 2

There's a nice paper by Kowalski, Robert, and Wu that discusses this problem. It's on the arxiv here.

Answer 3

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

1. $ \zeta_{2-k}$ is a differential operator
2. H = Harmonic Maass forms
3. 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