Decoupling, exponential sums and the Riemann zeta function

By Jean Bourgain. 2014

326 citations recorded by the source corpus.

We establish a new decoupling inequality for curves in the spirit of [B-D1], [B-D2] which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in [H]. In particular, this leads to an improved bound $|\zeta(\frac 12+it)|\ll t^{53/342+\varepsilon}$ for the zeta function on the critical line

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c107788e742e451436d1f21e42c85185601a32a9

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three (2015, unknown): “See [5], [6], [8],[7], [10] and [12].”
The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, background): “[85, Chapter V] or [19] for the currently best known bound μ0 = 13/84 + ε, due to Bourgain): for σ ≥ 12 , (2.”
On the Vinogradov mean value (2016, background): “hat there is a similar decoupling inequality for 2 ≤ p<k(k+ 1), though for k≥ 3 this is not just a consequence of interpolation. (iii) Our decoupling inequalities for curves appear in [B1], [BD2], [B2]. In particular, the reader is referred to [B2] for an application to exponential sums and the L”
On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, background): “If one does not assume the Riemann hypothesis, the best known result in this direction, due to Bourgain [9], is that the estimate is true for all α > 13/84.”
Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, background): “x;x+ x1=2+"] for all xlarge enough, and in almost all intervals of the form [x;x+ x"]. The current best bound towards the Lindelof hypothesis states that j(1 2 + it)j˝jtj13=84+" (see [8]). Chapter XIII of [41] gives a more comprehensive account of the literature surrounding the Lindel”