Book ChapterDOI
The Sato–Tate Conjecture for the Ramanujan τ-Function
M. Ram Murty,V. Kumar Murty +1 more
- pp 155-171
TLDR
In this paper, the authors give a sketch of how their proof works, and also indicate some lines of future development, and show that the semi-circular conjecture can be proved.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.read more
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