Odd values of the Ramanujan tau function

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 eitherlog log log n log log log log n 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 b 1 5 b 2 7 b 3 11 b 4 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.Communicated by Kannan Soundararajan. Michael A. Bennett is supported by NSERC. Adela Gherga and Samir Siksek are supported by an EPSRC Grant EP/S031537/1 "Moduli of elliptic curves and classical Diophantine problems".

Even values of Ramanujan's tau-function

“…Remark. As noted earlier, the present work was nearly finished when preprint [8] was posted on the arXiv. Thanks to their work, we note that the claims in Theorem 1.1 and Corollary 1.2 that rely on GRH are unconditionally true.…”

“…Recently, 1 Bennett, Gherga, Patel, and Siksek [8] proved a number of spectacular results regarding odd values of τ (n). If P (m) denotes the largest prime factor of m, they prove the existence of an effectively computable constant κ such that for odd τ (n) with n ≥ 25, then either…”

“…Thanks to their work, we note that the claims in Theorem 1.1 and Corollary 1.2 that rely on GRH are unconditionally true. To see this, one applies Theorem 6 of [8], which implies for odd primes ℓ < 100 that |τ (n)| = ℓ 2 , in the proof of Case We note that 277 and 1297 are both prime. Every such value with n ≤ 200, 000 has prime n.…”

Some Remarks on Small Values of $τ(n)$

“…Since none of the values of B listed in Table 1 are 11th powers, the sporadic examples listed here do not apply to the Lucas sequence u n . Now, by the results of Lygeros and Rozier [13] as well as Bennett, Gherga, Patel, and Siksek [5] stated in Section 1, we have that τ (p a ) = −1, ±3 r as in rows 1 and 2 of Table 2. Rows 3, 5, and 7 give rise to a defective term only if m = τ (p) is odd; however, by (1.2) we know that τ (p) is even for all primes.…”

“…In 2020, Hanada and Madhukara [11] additionally proved that τ (n) / ∈ {−9, ±15, ±21, −25, −27, −33, ±35, ±45, ±49, −55, ±63, ±77, −81, ±91}, and Dembner and Jain [9] showed that τ (n) = ±ℓ, where ℓ < 100 is any odd prime. Shortly thereafter, Bennett, Gherga, Patel, and Siksek [5] proved that the same is true for any positive power of ℓ, and that τ (n) = ±3 a 5 b 7 c 11 d for any a, b, c, d ∈ Z ≥0 and any n > 1. Using some of these results concerning odd primes, Balakrishnan, Ono, and Tsai [4] were able to make progress eliminating even integers as values of the τ -function.…”