Odd values of the Ramanujan tau function

2021/01/08 by Bennett, Michael, Gherga, Adela, Patel, Vandita +1
#11D61 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2101.02933

Abstract

We prove a number of results regarding odd values of the Ramanujan τ-function. For example, we prove the existence of an effectively computable positive constant κ such that if τ(n) is odd and n ≥ 25 then either P(τ(n)) gt; κ⋅ \fraclogloglognloglogloglogn or there exists a prime p | n with τ(p)=0. Here P(m) denotes the largest prime factor of m. We also solve the equation τ(n)=± 3^b₁ 5^b₂ 7^b₃ 11^b₄ and the equations τ(n)=± q^b where 3≤ 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 Győry, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves.

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