On some values which do not belong to the image of Ramanujan's tau-function

2024/05/26 by Goto, Akihiro
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2405.16723

Abstract

Lehmer conjectured that Ramanujan's tau function never vanishes. As a variation of this conjecture, it is proved that τ(n)≠ ± ℓ, ± 2ℓ, ± 2ℓ², where ℓ<100 is an odd prime, by Balakrishnan, Ono, Craig, Tsai and many people. We have proved that τ(n)≠ ± ℓ, ± 2ℓ, ± 4ℓ, ± 8ℓ for any n≥ 1 except 14 cases, where ℓ<1000 is an odd prime.

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• Variations of Lehmer's Conjecture for Ramanujan's tau-function
• Odd values of the Ramanujan tau function
• Lehmer's Conjecture on the Non-vanishing of Ramanujan's Tau Function
• Elementary Proof of Lehmer's Conjecture on Non-vanishing of Tau Function
• Prime values of Ramanujan's tau function