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 $\tau (n)\ne \alpha$ 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

$$\begin{aligned} \tau (n)\not \in \{ \pm 2\ell \ : \ 3\le \ell< 100\}&\cup&\{\pm 2\ell ^2 \ : \ 3\le \ell<100\}\\&\cup&\{\pm 2\ell ^3 \ : \ 3\le \ell <100\ {\text {with }\ell \ne 59}\}. \end{aligned}$$

Moreover, we obtain such results for infinitely many powers of each prime $3\le \ell <100$. As an example, for $\ell =97$ we prove that

$$\begin{aligned} \tau (n)\not \in \{ 2\cdot 97^j \ : \ 1\le j\not \equiv 0\pmod {44}\}\cup \{-2\cdot 97^j \ : \ j\ge 1\}. \end{aligned}$$

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.