Tau function: Difference between revisions
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In [[mathematics]], Ramanujan's '''tau function''' is an [[arithmetic function]] which may defined in terms of the [[Delta form]] by the formal infinite product | In [[mathematics]], [[Srinivasa Ramanujan]]'s '''tau function''' is an [[arithmetic function]] which may defined in terms of the [[Delta form]] by the formal infinite product | ||
:<math>q \prod_{n=1}^\infty \left(1-q^n\right)^{24} = \sum_n \tau(n) q^n .\,</math> | :<math>q \prod_{n=1}^\infty \left(1-q^n\right)^{24} = \sum_n \tau(n) q^n .\,</math> | ||
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:<math> \sum_n \tau(n) n^{-s} = \prod_p \left(1 - \tau(p) p^{-s} + p^{11-2s} \right)^{-1} .\,</math> | :<math> \sum_n \tau(n) n^{-s} = \prod_p \left(1 - \tau(p) p^{-s} + p^{11-2s} \right)^{-1} .\,</math> | ||
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Latest revision as of 07:01, 25 October 2024
In mathematics, Srinivasa Ramanujan's tau function is an arithmetic function which may defined in terms of the Delta form by the formal infinite product
Since Δ is a Hecke eigenform, the tau function is multiplicative, with formal Dirichlet series and Euler product
