Even Values of Ramanujan’s Tau-Function

• Published in: La matematica Vol. 1; no. 2; pp. 395 - 403
• Main Authors: Balakrishnan, Jennifer S., Ono, Ken, Tsai, Wei-Lun
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2022
• Subjects: Mathematics; Mathematics and Statistics; Original Research Article; Modular forms; Newforms; Lehmer’s conjecture; Ramanujan’s tau-function
• ISSN: 2730-9657; 2730-9657
• DOI: 10.1007/s44007-021-00005-8

In the spirit of Lehmer’s speculation that Ramanujan’s tau-function never vanishes, it is natural to ask whether any given integer α is a value of τ ( n ) . For odd α , Murty, Murty, and Shorey proved that τ ( n ) ≠ α for sufficiently large n . Several recent papers have identified explicit examples of odd α which are not tau-values. Here we apply these results (most notably the recent work of Bennett, Gherga, Patel, and Siksek) to offer the first examples of even integers that are not tau-values. Namely, for primes ℓ we find that τ ( n ) ∉ { ± 2 ℓ : 3 ≤ ℓ < 100 } ∪ { ± 2 ℓ 2 : 3 ≤ ℓ < 100 } ∪ { ± 2 ℓ 3 : 3 ≤ ℓ < 100 with ℓ ≠ 59 } . Moreover, we obtain such results for infinitely many powers of each prime 3 ≤ ℓ < 100 . As an example, for ℓ = 97 we prove that τ ( n ) ∉ { 2 · 97 j : 1 ≤ j ≢ 0 ( mod 44 ) } ∪ { - 2 · 97 j : j ≥ 1 } . The method of proof applies mutatis mutandis to newforms with residually reducible mod 2 Galois representation and is easily adapted to generic newforms with integer coefficients.