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The Sato–Tate Conjecture for the Ramanujan τ-Function

In: The Mathematical Legacy of Srinivasa Ramanujan

Author

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  • M. Ram Murty

    (Queen’s University, Department of Mathematics and Statistics)

  • V. Kumar Murty

    (University of Toronto, Department of Mathematics)

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.

Suggested Citation

  • M. Ram Murty & V. Kumar Murty, 2013. "The Sato–Tate Conjecture for the Ramanujan τ-Function," Springer Books, in: The Mathematical Legacy of Srinivasa Ramanujan, edition 127, chapter 0, pages 155-171, Springer.
    • Handle: RePEc:spr:sprchp:978-81-322-0770-2_12
    DOI: 10.1007/978-81-322-0770-2_12
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