Small Primes And The Tau Function

TL;DR

This paper demonstrates that the Ramanujan tau function does not take prime values for any prime less than or equal to 8.0×10^{25}, revealing a significant gap in its prime outputs.

Contribution

It establishes a new large bound showing the tau function's prime values do not occur below this threshold, advancing understanding of its prime value distribution.

Findings

Tau function misses all primes ≤ 8.0×10^{25}

No prime values of tau function below this bound

Provides a new large-scale computational result

Abstract

This note shows that the prime values of the Ramanujan tau function $\tau(n)=\pm p$ misses every prime $p\leq 8.0\times 10^{25}$.