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Prime values of Ramanujan's tau function

Boyuan Xiong

TL;DR

The paper investigates how often primes occur as values of Ramanujan's $\tau$-function. It combines Hecke multiplicativity, modular congruences modulo $23$, and Diophantine tools (Schinzel-type bounds and Bombieri–van der Poorten estimates) to control $\tau(p^k)$ for prime powers. It proves that for many residue classes mod $23$, the primes that are $\tau$-values are extremely sparse, and in particular there exists a set of primes of Dirichlet density at least $\frac{9}{11}$ that are not $\tau$-values, while the set of primes that are $\tau$-values has density zero. The results complement questions around Lehmer's conjecture by focusing on the image of $\tau$ rather than its zeroes, and showcase a sharp dichotomy between prime values and omitted values in the $\tau$-orbit.

Abstract

We study the prime values of Ramanujan's tau function $τ(n)$. Lehmer found that $n=251^2=63001$ is the smallest $n$ such that $τ(n)$ is prime: $$τ(251^2)=-80561663527802406257321747.$$ We prove that in most arithmetic progressions (mod 23), the prime values $τ$ belonging to the progression form a thin set. As a consequence, there exists a set of primes of Dirichlet density $\frac{9}{11}$ which are not values of $τ$.

Prime values of Ramanujan's tau function

TL;DR

The paper investigates how often primes occur as values of Ramanujan's -function. It combines Hecke multiplicativity, modular congruences modulo , and Diophantine tools (Schinzel-type bounds and Bombieri–van der Poorten estimates) to control for prime powers. It proves that for many residue classes mod , the primes that are -values are extremely sparse, and in particular there exists a set of primes of Dirichlet density at least that are not -values, while the set of primes that are -values has density zero. The results complement questions around Lehmer's conjecture by focusing on the image of rather than its zeroes, and showcase a sharp dichotomy between prime values and omitted values in the -orbit.

Abstract

We study the prime values of Ramanujan's tau function . Lehmer found that is the smallest such that is prime: We prove that in most arithmetic progressions (mod 23), the prime values belonging to the progression form a thin set. As a consequence, there exists a set of primes of Dirichlet density which are not values of .
Paper Structure (6 sections, 15 theorems, 54 equations)

This paper contains 6 sections, 15 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Main idea
  3. When $k$ is small
  4. When $k$ is large
  5. When $k$ is not large
  6. Proof of Theorem \ref{['T']}

Key Result

Proposition 1.1

Suppose that $l$ is an odd prime such that $l \nmid \tau(l)$. If $\tau(n) = \pm l^m$, with $m \in \mathbb{Z}^+$, then $n = p^{d-1}$, where $p$ and $d \text{ }|\text{ } l(l^2-1)$ are odd primes. Furthermore, $\tau(n) = \pm l^m$ for at most finitely many $n$.

Theorems & Definitions (16)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Corollary 3.2
  • Proposition 4.1
  • Corollary 4.2
  • ...and 6 more