The Sato–Tate Conjecture for the Ramanujan τ-Function

M. Ram Murty and V. Kumar Murty
Additional contact information
M. Ram Murty: Queen’s University, Department of Mathematics and Statistics
V. Kumar Murty: University of Toronto, Department of Mathematics

Chapter Chapter 12 in The Mathematical Legacy of Srinivasa Ramanujan, 2013, pp 155-171 from Springer

Abstract: 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.

Keywords: Elliptic Curve; Elliptic Curf; Riemann Hypothesis; Automorphic Representation; Prime Number Theorem (search for similar items in EconPapers)
Date: 2013
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-81-322-0770-2_12

Ordering information: This item can be ordered from
http://www.springer.com/9788132207702

DOI: 10.1007/978-81-322-0770-2_12

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().