Odd values of the Ramanujan tau function

TL;DR

This paper investigates the properties of odd values of the Ramanujan tau function, establishing bounds, solving specific equations, and employing advanced number theory techniques to deepen understanding of its behavior.

Contribution

It provides new bounds on odd tau values, solves specific exponential equations involving tau, and applies diverse modern number theory methods to analyze the function.

Findings

Existence of an effectively computable constant  for odd tau values

Solutions to ^{b_1} 5^{b_2} 7^{b_3} 11^{b_4} equations involving tau

Solutions to au(n)=\u00b1 q^b for primes q < 100

Abstract

We prove a number of results regarding odd values of the Ramanujan $\tau$-function. For example, we prove the existence of an effectively computable positive constant $\kappa$ such that if $\tau(n)$ is odd and $n \ge 25$ then either \[ P(\tau(n)) \; > \; \kappa \cdot \frac{\log\log\log{n}}{\log\log\log\log{n}} \] or there exists a prime $p \mid n$ with $\tau(p)=0$. Here $P(m)$ denotes the largest prime factor of $m$. We also solve the equation $\tau(n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$ and the equations $\tau(n)=\pm q^b$ where $3\le q < 100$ is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue--Mahler equations due to Bugeaud and Gy\H{o}ry, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves.