Mesoscopic fluctuations of the zeta zeros

By P. Bourgade. 2009

66 citations recorded by the source corpus.

We prove a multidimensional extension of Selberg’s central limit theorem for log ζ, in which non-trivial correlations appear. In particular, this answers a question by Coram and Diaconis about the mesoscopic fluctuations of the zeros of the Riemann zeta function. Similar results are given in the context of random matrices from the unitary group. This shows the correspondence n ↔ log t not only between the dimension of the matrix and the height on the critical line, but also, in a local scale, for small deviations from the critical axis or the unit circle.

Subfields: Algebra and Number Theory, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/20edf038acf14ef88f2ff51e701fb207d99c83ce

How researchers see this paper

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In Algebra and Number Theory

1. Joseph Najnudel
2. L. Arguin
3. P. Bourgade
4. Maksym Radziwill
5. David Belius
6. Christian Webb
7. Eero Saksman
8. Kannan Soundararajan

In Atomic and Molecular Physics, and Optics

1. Joseph Najnudel
2. David Belius
3. L. Arguin
4. Christian Webb
5. Eero Saksman

In Statistical and Nonlinear Physics

1. Joseph Najnudel
2. David Belius
3. L. Arguin
4. Christian Webb
5. Eero Saksman
6. P. Bourgade
7. Kannan Soundararajan

Citations in the corpus

• Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, background): “1 of [9], or Lemma 3 of [40]), and will be useful in comparing moments of Dirichlet polynomials over the primes with the moments of suitable Gaussian distributions.”
• On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, background): “This conjecture is not fully proven, but Chhaibi, Madaule and Najnudel in [13] have recently shown that the family (1.2) of random variables is tight, improving successive results by Arguin, Belius and Bourgade [3] and by Paquette and Zeitouni [26].”
• The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, background): “In the spirit of Selberg’s limit theorem, a partial result in the direction of a functional limit theorem for the logarithm of the zeta function on a mesoscopic scale was proven by Bourgade [17], the limiting object being a suitable Gaussian process (see also [48] for a related result).”
• Maxima of a randomized Riemann zeta function, and branching random walks (2015, unknown): “Bourgade showed a multivariate version of Selberg’s theorem where the correlations are logarithmic in the limit [9].”