Zeev Rudnick
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Zeev Rudnick has a total of 39 co-authors. Here are the top co-authors by number of shared publications:
- Jean Bourgain
Institute for Advanced Study
9 shared publications - Igor Wigman
Department of Mathematics, King’s College London, London, UK
6 shared publications - Bingrong Huang
Data Science Institute & State Key Laboratory of Cryptography and Digital Economy Security, Shandong University, Jinan, P. R. China
3 shared publications - Henrik Ueberschaer
Tel Aviv University
3 shared publications - Jonathan P. Keating
Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
3 shared publications - C. P. Hughes2 shared publications
Latest Publications
- 30 April 2019
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 232 (1) , 159-199
Angles of Gaussian primes
- 30 April 2019
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 232 (1) , 159-199
Angles of Gaussian primes
- Open Access
Gaussian primes in almost all narrow sectors
We study Gaussian primes lying in narrow sectors, and show that almost all such sectors contain the expected number of primes, if the sectors are not too narrow.
Open AccessGaussian primes in almost all narrow sectors
We study Gaussian primes lying in narrow sectors, and show that almost all such sectors contain the expected number of primes, if the sectors are not too narrow.
- Open Access
Prime lattice points in ovals
We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term.
Open AccessPrime lattice points in ovals
We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term.
- 22 October 2018
- Published by Springer Nature in Monatshefte für Mathematik
- Vol. 189 (2) , 295-319
Prime lattice points in ovals
- 22 October 2018
- Published by Springer Nature in Monatshefte für Mathematik
- Vol. 189 (2) , 295-319
Prime lattice points in ovals
- Open Access
Angles of Gaussian primes
Fermat showed that every prime p = 1 mod 4 is a sum of two squares: $p = a^2 + b^2$. To any of the 8 possible representations (a,b) we associate an angle whose tangent is the ratio b/a. In 1919 Hecke showed that these angles are uniformly distributed as p varies, and in the 1950's Kubilius proved uniform distribution in somewhat short arcs. We study fine scale statistics of these angles, in particular the variance of the number of such angles in a short arc. We present a conjecture for this variance, motivated both by a random matrix model, and by a function field analogue of this problem, for which we prove an asymptotic form for the corresponding variance.Open AccessAngles of Gaussian primes
Fermat showed that every prime p = 1 mod 4 is a sum of two squares: $p = a^2 + b^2$. To any of the 8 possible representations (a,b) we associate an angle whose tangent is the ratio b/a. In 1919 Hecke showed that these angles are uniformly distributed as p varies, and in the 1950's Kubilius proved uniform distribution in somewhat short arcs. We study fine scale statistics of these angles, in particular the variance of the number of such angles in a short arc. We present a conjecture for this variance, motivated both by a random matrix model, and by a function field analogue of this problem, for which we prove an asymptotic form for the corresponding variance. - Open Access
Points on nodal lines with given direction
We study of the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. We give upper bounds for the flat torus, and compute the expected number for arithmetic random waves.
Open AccessPoints on nodal lines with given direction
We study of the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. We give upper bounds for the flat torus, and compute the expected number for arithmetic random waves.
- 1 January 2018
- Published by Wiley in Mathematika
- Vol. 64 (3) , 628-636
A METRIC THEORY OF MINIMAL GAPS
- 1 January 2018
- Published by Wiley in Mathematika
- Vol. 64 (3) , 628-636
A METRIC THEORY OF MINIMAL GAPS
- Open Access
A metric theory of minimal gaps
We study the minimal gap statistic for fractional parts of sequences of the form $\mathcal A^\alpha = \{\alpha a(n)\}$ where $\mathcal A = \{a(n)\}$ is a sequence of distinct of integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\alpha$, the minimal gap $\delta_{\min}^\alpha(N)=\min\{\alpha a(m)-\alpha a(n)\bmod 1: 1\leq m\neq n\leq N\}$ is close to that of a random sequence.Open AccessA metric theory of minimal gaps
We study the minimal gap statistic for fractional parts of sequences of the form $\mathcal A^\alpha = \{\alpha a(n)\}$ where $\mathcal A = \{a(n)\}$ is a sequence of distinct of integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\alpha$, the minimal gap $\delta_{\min}^\alpha(N)=\min\{\alpha a(m)-\alpha a(n)\bmod 1: 1\leq m\neq n\leq N\}$ is close to that of a random sequence. - Open Access
Small gaps in the spectrum of the rectangular billiard
We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $\alpha$, in comparison to the corresponding quantity for a Poissonian sequence. If $\alpha$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of $\alpha$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $\alpha$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory.Open AccessSmall gaps in the spectrum of the rectangular billiard
We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $\alpha$, in comparison to the corresponding quantity for a Poissonian sequence. If $\alpha$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of $\alpha$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $\alpha$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory. - 9 August 2016
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 350 (1) , 279-300
Small Scale Equidistribution of Eigenfunctions on the Torus
- 9 August 2016
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 350 (1) , 279-300
Small Scale Equidistribution of Eigenfunctions on the Torus