# Known and unknown about Ramanujan's tau function

What is a good reference for open problems relating to the Ramanujan tau function?

I know about Lehmer's conjecture. I know the following reductions of the problem: the smallest counterexample must be a power of a prime (by the multiplicativity of $\tau$), it must be a prime (by considering the linear recurrence expressing $\tau(p^{n})$ in terms of $\tau(p^{n-1})$ and $\tau(p^{n-2})$ ), and it must be a prime $p$ such that $p \equiv -1 \mod{691}$ (this follows from the congruence $\sigma_{11}(n) \equiv \tau(n) \mod{691}$). Are all the other reductions, like the last one I mentioned, just obtained from congruences satisfied by $\tau$?

I have read that problems about the sign of $\tau$ tend to be hard, but I don't know precisely about this (aside from the problem I just mentioned about the vanishing of $\tau$). Where would problems about the sign of $\tau$ be discussed in detail?

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The smallest $p$ for which $\tau(p)=0$ (assuming there is such a $p$) is now known to be huge, simply because of congruence reasons (we know what it must be congruent to modulo a big power of 2, a biggish power of 3, and then other primes like $\ell=5$, 7, 23, 691 (where the mod $\ell$ rep is degenerate), and then more recently primes like 11 too, even though the mod 11 representation is irreducible). As for signs, by Sato-Tate (which is known for $\Delta$ now) it's 50-50, and it's not clear to me what other problems are left because the sign is surely random in some sense. – user30035 Mar 16 '13 at 14:33
Well, I've never had any answers to this question: mathoverflow.net/questions/38691/… – Stopple Mar 16 '13 at 17:43
Stopple: I don't think that that question is a question about the Delta function, it's a question about a few numbers which happen to be related to the Delta function, and I think that the answer to that question is that if you do enough random calculations of this nature then of course you'll find coincidences like that -- there are several questions on this site of the form "look at this 3-digit number showing up in two places -- is this a coincidence?" and the answer in most cases is "yes". – user30035 Mar 16 '13 at 17:47
You mean, it's 50-50 for those $n$ such that $\tau(n) \neq 0$. If there is a prime $p$ such that $\tau(p) = 0$, then multiplicativity of $\tau$ yields $\tau(pm) = 0$ whenever $m$ is a nonmultiple of $p$, giving a set of density $\frac{1}{p} - \frac{1}{p^{2}}$ on which $\tau$ vanishes. Thanks for the update on Sato-Tate. – DavidLHarden Mar 17 '13 at 7:44
The mod $\l$ representation of what group is degenerate modulo 691? 691 doesn't divide the order of any Conway group -- indeed, no prime exceeding 71 divides the order of a sporadic group. – DavidLHarden Mar 20 '13 at 19:23