Even values of Ramanujan's tau-function

Abstract

In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer $\alpha$ is a value of $\tau(n)$. For odd $\alpha$, Murty, Murty, and Shorey proved that $\tau(n)\neq \alpha$ for sufficiently large $n$. Several recent papers have identified explicit examples of odd $\alpha$ which are not tau-values. Here we apply these results (most notably the recent work of Bennett, Gherga, Patel, and Siksek) to offer the first examples of even integers that are not tau-values. Namely, for primes $\ell$ we find that $$ \tau(n)\not \in \{ \pm 2\ell \ : \ 3\leq \ell< 100\} \cup \{\pm 2\ell^2 \ : \ 3\leq \ell <100\} \cup \{\pm 2\ell^3 \ : \ 3\leq \ell<100\ {\text {\rm with $\ell\neq 59$}}\}.$$ Moreover, we obtain such results for infinitely many powers of each prime $3\leq \ell<100$. As an example, for $\ell=97$ we prove that $$\tau(n)\not \in \{ 2\cdot 97^j \ : \ 1\leq j\not \equiv 0\pmod{44}\}\cup \{-2\cdot 97^j \ : \ j\geq 1\}.$$ The method of proof applies mutatis mutandis to newforms with residually reducible mod 2 Galois representation and is easily adapted to generic newforms with integer coefficients.