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 $$\tau (n)\notin \{\pm 2\ell :3\le \ell <100\}\cup \{\pm 2{\ell }^2:3\le \ell <100\}\cup \{\pm 2{\ell }^3:3\le \ell <100\text{with ℓ=59}\}.$$ 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 $$\tau (n)\notin \{2\cdot 97^j:1\le j\not\equiv 0(\mathrm{mod}44)\}\cup \{-2\cdot 97^j:j\ge 1\}.$$ 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.

La Matematica (2021)

Cited by in corpus (1)

• Polynomial interpolation of modular forms for Hecke groups