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The Sato–Tate Conjecture for the Ramanujan τ-Function

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The Mathematical Legacy of Srinivasa Ramanujan

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Abstract

Ramanujan’s 1916 conjecture that |τ(p)|≤2p 11/2 was proved in 1974 by P. Deligne, as a consequence of his work on the Weil conjectures. Serre, and later Langlands, discussed the possible distribution of the τ(p)/2p 11/2 in the interval [−1,1] as p varies over the prime numbers. Inspired by the Sato–Tate conjecture in the theory of elliptic curves, Serre predicted an identical distribution law (the “semi-circular” law). This conjecture was proved recently by Barnet-Lamb, Geraghty, Harris, and Taylor. In this chapter, we give a sketch of how their proof works. We also indicate some lines of future development.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada

    M. Ram Murty

  2. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada

    V. Kumar Murty

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  1. M. Ram Murty
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  2. V. Kumar Murty
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Murty, M.R., Murty, V.K. (2013). The Sato–Tate Conjecture for the Ramanujan τ-Function. In: The Mathematical Legacy of Srinivasa Ramanujan. Springer, India. https://doi.org/10.1007/978-81-322-0770-2_12

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