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Decoupling, exponential sums and the Riemann zeta function


Author: J. Bourgain
Journal: J. Amer. Math. Soc. 30 (2017), 205-224
MSC (2010): Primary 11M06, 11L07
DOI: https://doi.org/10.1090/jams/860
Published electronically: March 17, 2016
MathSciNet review: 3556291
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Abstract: We establish a new decoupling inequality for curves in the spirit of earlier work of C. Demeter and the author which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in the work of Huxley. In particular, this leads to an improved bound $ \vert\zeta (\frac {1}{2} + it)\vert \ll t^{13/84 + \varepsilon }$ for the zeta function on the critical line.


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Additional Information

J. Bourgain
Affiliation: Institute for Advanced Study, Princeton, New Jersey 08540
Email: bourgain@math.ias.edu

DOI: https://doi.org/10.1090/jams/860
Keywords: Riemann zeta function, exponential sums, decoupling
Received by editor(s): September 12, 2014
Received by editor(s) in revised form: October 4, 2015, November 5, 2015, December 31, 2015, and February 19, 2016
Published electronically: March 17, 2016
Additional Notes: The author was partially supported by NSF grant DMS-1301619
Article copyright: © Copyright 2016 American Mathematical Society

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