Publications
Zeev Rudnick

Preprints

  1. Zeev Rudnick. The Quantum Rabi model: Towards Braak's conjecture. arXiv:2311.12622 [math.SP]

  2. Moubariz Garaev, Zeev Rudnick, Igor Shparlinski. On a family of sparse exponential sums. arXiv:2309.11887 [math.NT]

  3. Zeev Rudnick and Igor Wigman. Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus. arXiv:2301.05964 [math.SP]

Papers in Journals

  1. Z. Rudnick Representation varieties of solvable groups Journal of Pure and Applied Algebra 45, 261-272 (1987).

  2. Z. Rudnick, H. Schlichtkrull Decay of eigenfunctions on semisimple symmetric spaces. Duke Mathematical Journal 64, 445-450 (1991).

  3. A. Eskin, Z. Rudnick, P. Sarnak A proof of Siegel's weight formula , Duke Mathematical Journal International Math. Research Notes 64, 65-69 (1991).

  4. W. Duke, Z. Rudnick, P. Sarnak Density of integer points on affine homogeneous varieties , Duke Mathematical Journal 71, 143-179 (1993).

  5. R. Phillips, Z. Rudnick The circle problem in the hyperbolic plane , Jour. of Func. Anal. 121, 78-116 (1994).

  6. Z. Rudnick, P. Sarnak The behaviour of eigenstates of arithmetic hyperbolic manifolds , Comm. in Math. Physics 161, 195-213 (1994).

  7. Z. Rudnick Representations of compact p-adic groups, Appendix to Betti Numbers of Congruence Subgroups, by S. Adams and P. Sarnak, Israel Journal of Mathematics 88, 31--72 (1994).

  8. M. Borovoi, Z. Rudnick Hardy-Littlewood varieties and semisimple groups, Inventiones Math 119 , 37--66 (1995).

  9. Z. Rudnick, P. Sarnak The n-level correlations of zeros of the zeta function , C.R. Acad.Sci. Paris 319, 1027--1032 (1994).

  10. W. Luo, Z. Rudnick and P. Sarnak On Selberg's eigenvalue conjecture , Geom. and Func. Analysis 5 (1995), 387--401.

  11. Z. Rudnick and P. Sarnak Zeros of principal L-functions and random matrix theory , Duke Jour. of Math. 81 (1996), 269--322 (special volume in honor of J. Nash).

  12. D. Ginzburg, Z. Rudnick Stable multiplicities in the length spectrum of Riemann surfaces Israel Jour. of Math. 104 (1998), 129--144.

  13. Z. Rudnick and P. Sarnak The pair correlation function of fractional parts of polynomials Comm. in Math. Physics. 194 (1998), 61--70.

  14. W. Luo, Z. Rudnick, P. Sarnak On the generalized Ramanujan conjectures for GL(n) Proc. Symposia in Pure Math., volume 66 part 2, 301-310, American Math. Soc. 1999.

  15. D. Jakobson, S. Miller, I. Rivin and Z. Rudnick Level spacings for regular graphs in Emerging Applications of Number Theory, edited by D. Hejhal, J. Friedman, M. Gutzwiller and A. Odlyzko, IMA volumes in Math. and its Applications 109, Springer 1999

  16. Z. Rudnick and A. Zaharescu A metric result on the pair correlation of fractional parts of sequences , Acta Arithmetica LXXXIX (3) (1999), 283-293.

  17. P. Kurlberg and Z. Rudnick The distribution of spacings between quadratic residues Duke Jour. of Math. 100 (1999), 211-242.

  18. P. Kurlberg and Z. Rudnick Hecke theory and equidistribution for the quantization of linear maps of the torus , Duke Math. Jour. 103 (2000), 47--78.

  19. Z. Rudnick The number of solutions of simultaneous quadratic equations, Appendix to The n-point correlations between values of a linear form, by Jens Marklof, Ergodic Theory and Dynamical Systems 20 (2000) 1127-1172.

  20. Z. Rudnick and A. Zaharescu The distribution of spacings between small powers of a primitive root , Israel Jour. of Math. 120 Part A (2000), 271--287 .

  21. J. Marklof and Z. Rudnick Quantum unique ergodicity for parabolic maps , Geom. funct. anal. 10 Issue 6 (2000), pp 1554-1578

  22. Z. Rudnick, P. Sarnak and A. Zaharescu The distribution of spacings between the fractional parts of an 2 , Inventiones Math. 145 (2001), 37-57.

  23. P. Kurlberg and Z. Rudnick On quantum ergodicity for linear maps of the torus, Commun. Math. Phys. 222 (2001) 1, 201-227

  24. L. Polterovich and Z. Rudnick Kick stability in groups and dynamical systems , Nonlinearity 14 No. 5 (2001), 1331-1363.

  25. P. Kurberg and Z. Rudnick Value distribution for eigenfunctions of desymmetrized quantum maps , Internat. Math. Res. Notices 18 2001, 995-1002

  26. Z. Rudnick and A. Zaharescu The distribution of spacings between fractional parts of lacunary sequences , Forum Math. 14, 2002, 691-712.

  27. C.P. Hughes and Z. Rudnick Linear statistics for zeros of the Riemann zeta function , C. R. Acad. Sci. Paris, Ser I 335 (2002), 667-670. math.nt/0208220 ( long version).

  28. J.P. Keating, N.Linden and Z. Rudnick Random Matrix Theory, the exceptional Lie groups, and L-functions , J. Phys. A 36 no. 12 (special RMT volume) (2003), 2933-2944.

  29. C.P. Hughes and Z. Rudnick Mock Gaussian behaviour for linear statistics of classical compact groups , J. Phys. A 36 no. 12 (special RMT volume) (2003), 2919-2932.

  30. C.P. Hughes and Z. Rudnick Linear statistics of low-lying zeros of $L$--functions , Quarterly Jour. of Math. 54, Issue 3 (2003), 309-333.

  31. N. Ailon and Z. Rudnick Torsion points on curves and common divisors of ak - 1 and bk - 1 , Acta Arith. 113 (2004), no. 1, 31--38.

  32. L. Polterovich and Z. Rudnick Stable mixing for cat maps and quasi-morphisms of the modular group , Ergodic Theory and Dynamical systems. (2004), 24, 609-619

  33. C.P. Hughes and Z. Rudnick On the distribution of lattice points in thin annuli, IMRN 13 (2004), 637--658.

  34. Pietro Corvaja, Zeev Rudnick and Umberto Zannier A lower bound for periods of matrices, Comm. Math. Phys. 252 (2004), no. 1-3, 535--541. (Dyson volume).

  35. P. Kurlberg and Z. Rudnick On the distribution of matrix elements for the quantum cat map, Annals of Math. 161 (1) (2005), 489-507.

  36. Z. Rudnick A central limit theorem for the spectrum of the modular group (Park city lectures), Annales Henri Poincare 6 (2005), 863-883.

  37. Z. Rudnick and K. Soundararajan Lower bounds for moments of L-functions , Proc. of the National Academy of Sciences of the USA, 102 (19), (May 10, 2005), 6837-6838.

  38. Z. Rudnick On the asymptotic distribution of zeros of modular forms , IMRN number 34 (2005), 2059--2076.

  39. Z. Rudnick and K. Soundararajan, Lower bounds for moments of L-functions: symplectic and orthogonal examples, in proceedings of the Bretton Woods Workshop on Multiple Dirichlet Series, Proceedings of Symposia in Pure Mathematics 75, the American Mathematical Society (2006).

  40. P. Kurlberg, L. Rosenzweig, Z. Rudnick, Matrix elements for the quantum cat map: Fluctuations in short windows , Nonlinearity 20 (2007), 2289-2304.

  41. Ferenc Oravecz, Zeev Rudnick and Igor Wigman, The Leray measure of nodal sets for random eigenfunctions on the torus, Annales de l'Institut Fourier 58 (1), (2008), 299-335.

  42. Zeev Rudnick and Igor Wigman, On the volume of nodal sets for eigenfunctions of the Laplacian on the torus, Annales Henri Poincare 9 (2008), 109-130.

  43. Morten S. Risager and Zeev Rudnick Equidistribution of the real part of orbits in hyperbolic spaces , in proceedings of the conference on the occasion of Sunada's 60th birthday (Nagoya 2007). Contemp.Math. series of Amer. Math.Soc.

  44. P. Kurlberg, Z. Rudnick The fluctuations in the number of points on a hyperelliptic curve over a finite field , Journal of Number Theory 129 (2009), 580--587.

  45. Dmitry Faifman, Zeev Rudnick Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field , Compositio Math. 146 (2010), 81--101.

  46. Wenzhi Luo, Zeev Rudnick and Peter Sarnak The variance of arithmetic measures associated to closed geodesics on the modular surface , Jour. of Modern Dynamics, volume 3 no. 2, 2009, 271--309 .

  47. J. Bourgain and Z. Rudnick Restriction of toral eigenfunctions to hypersurfaces, C.R. Math. Acad. Sci. Paris 347 (2009), no 21--22, 1249--1253.

  48. Zeev Rudnick Traces of high powers of the Frobenius class in the hyperelliptic ensemble , Acta Arith. 143 (2010), 81--99.

  49. Jean Bourgain and Zeev Rudnick On the nodal sets of toral eigenfunctions , Inventiones Math. Volume 185 Number 1 (2011), 199-237.

  50. Jean Bourgain and Zeev Rudnick On the geometry of the nodal lines of eigenfunctions on the two-dimensional torus , Ann. Henri Poincare 12 (2011), no. 6, 1027--1053. AHP Distinguished Paper Award for the year 2011.

  51. Jean Bourgain and Zeev Rudnick Restriction of toral eigenfunctions to hypersurfaces and nodal sets , Geometric and Functional Analysis: Volume 22, Issue 4 (2012), Page 878-937.

  52. Jens Marklof and Zeev Rudnick Almost all eigenfunctions of a rational polygon are uniformly distributed. J. of Spectral Theory. 2 (2012), 107--113.

  53. Zeev Rudnick and Henrik Ueberschaer Wave function statistics for a point scatterer on the torus. , Comm. Math. Phys. 316, Issue 3 (2012), 763--782.

  54. Alexei Entin, Edva Roditty-Gershon and Zeev Rudnick, Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and Random Matrix Theory. , Geometric and Functional Analysis Volume 23 Issue 4 (2013), 1230--1261. doi:10.1007/s00039-013-0241-8.

  55. Jonathan P. Keating and Zeev Rudnick The variance of the number of prime polynomials in short intervals and in residue classes. Int. Math. Res. Not. IMRN 2014, no. 1, 259-288.

  56. Dan Carmon and Zeev Rudnick The autocorrelation of the Mobius function and Chowla's conjecture for the rational function field, Q. J. Math. 65 (2014), no. 1, 53-61.

  57. Zeev Rudnick and Henrik Ueberschaer, On the eigenvalue spacing distribution for a point scatterer on the flat torus , Ann. Henri Poincare 15 (2014), no. 1, 1-27.

  58. Z. Rudnick Square-free values of polynomials over the rational function field , Journal of Number Theory, Volume 135, February 2014, Pages 60--66. DOI: 10.1016/j.jnt.2013.08.014

  59. Jean Bourgain and Zeev Rudnick, Nodal intersections and Lp restriction theorems on the torus. , ISRAEL JOURNAL OF MATHEMATICS 207 (2015), 479--505. DOI 10.1007/s11856-015-1183-7

  60. J. C. Andrade, L. Bary-Soroker, and Z. Rudnick Shifted convolution and the Titchmarsh divisor problem over Fq[t]. , Number fields and function fields: coalescences, contrasts and emerging applications. Phil. Trans. R. Soc. A 373: 20140315. (2015). See erratum correcting a misprint here .

  61. Jean Bourgain, Zeev Rudnick and Peter Sarnak Local statistics of lattice points on the sphere. , Contemporary Mathematics vol. 616, (2016), 269-282 (proceedings of Constructive Functions 2014).

  62. J.P. Keating and Z. Rudnick, Squarefree polynomials and Mobius values in short intervals and arithmetic progressions , Algebra & Number Theory vol 10 (no. 2) 2016, 375-420.

  63. Z. Rudnick and I. Wigman Nodal intersections for random eigenfunctions on the torus. , Amer. J. of Math. Volume 138, Number 6, December 2016 pp. 1605--1644.

  64. Z. Rudnick, I. Wigman and N. Yesha Nodal intersections for random waves on the 3-dimensional torus. , Ann. Inst. Fourier (Grenoble) 66 (2016), no. 6, 2455--2484.

  65. Steve Lester and Zeev Rudnick Small scale equidistribution of eigenfunctions on the torus. , Comm. Math. Phys. 350 (2017), no. 1, 279--300.

  66. Zeev Rudnick and Xin Zhang Gap distributions in circle packings , Munster J. Math. 10 (2017), no. 1, 131–170.

  67. Jean Bourgain, Zeev Rudnick and, Peter Sarnak, Spatial statistics for lattice points on the sphere I: Individual results. , Bull. Iranian Math. Soc. 43 (2017), no. 4, 361–386. special issue in honor of Freydoon Shahidi's 70'th birthday.

  68. Valentin Blomer, Jean Bourgain, Maksym Radziwill and Zeev Rudnick, Small gaps in the spectrum of the rectangular billiard. , Annales Scientifiques de l'Ecole Normale Superieure, 50, fascicule 5 (2017), p. 1283--1300.

  69. Jon Keating, Brad Rodgers, Edva Roditty-Gershon, and Zeev Rudnick Sums of divisor functions in Fq[t] and matrix integrals. , Math. Z. 288 (2018), no. 1-2, 167–198.

  70. Zeev Rudnick A metric theory of minimal gaps , Mathematika. Volume 64, Issue 3 2018 , pp. 628-636

  71. Bingrong Huang and Zeev Rudnick Prime lattice points in ovals . Monatshefte für Mathematik (2019) 189:295–319

  72. Zeev Rudnick On locally repeated values of arithmetic functions over Fq[T], with an appendix by Ron Peled. Q. J. Math. 70 (2019), no. 2, 451–472.

  73. Zeev Rudnick and Ezra Waxman, Angles of Gaussian primes. Israel J. Math. 232 (2019), no. 1, 159–199.

  74. Bingrong Huang, Jianya Liu and Zeev Rudnick Gaussian primes in almost all narrow sectors . Acta Arithmetica 193 (2020), 183-192. DOI: 10.4064/aa190331-23-7

  75. Zeev Rudnick and Igor Wigman Points on nodal lines with given direction . J. of Spectral Theory. Volume 10, Issue 3, 2020, pp. 927–953. DOI: 10.4171/JST/317

  76. Tom Dvir, Renana Peres and Zeev Rudnick Modelling the expected probability of correct assignment under uncertainty . Scientific Reports 10, 15080 (2020).

  77. Zeev Rudnick and Niclas Technau, The metric theory of the pair correlation function of real-valued lacunary sequences . Illinois Journal of Mathematics, Vol. 64, No. 4 (2020), 583–594. DOI 10.1215/00192082-8720506.

  78. James Maynard and Zeev Rudnick, A lower bound on the LCM of polynomial sequences. Riv. Mat. Univ. Parma, Vol. 12, No. 1, 2021, 143--150.

  79. Zeev Rudnick and Sa'ar Zehavi On Cilleruelo's conjecture for the least common multiple of polynomial sequences . Revista Matematica Iberoamericana. Volume 37, Issue 4, 2021, pp. 1441--1458 DOI: 10.4171/rmi/1234

  80. Zeev Rudnick and Igor Wigman On the Robin spectrum for the hemisphere . arXiv:2008.12964 [math.SP]. Annales Mathématiques du Québec (special issue in honor of A. Shnirelman), doi.org/10.1007/s40316-021-00155-9.

  81. Wen-Ching Winnie Li and Zeev Rudnick Pair arithmetical equivalence for quadratic fields. Math. Z. 299, 797–826 (2021). https://doi.org/10.1007/s00209-021-02706-w

  82. Zeev Rudnick, Igor Wigman, Nadav Yesha Differences between Robin and Neumann eigenvalues . Commun. Math. Phys. 388, 1603–1635 (2021). https://doi.org/10.1007/s00220-021-04248-y

  83. Zeev Rudnick and Igor Wigman The Robin problem on rectangles. J. Math. Phys. 62, 113503 (2021); https://doi.org/10.1063/5.0061763, special collection of papers honoring Freeman Dyson.

  84. Zeev Rudnick and Igor Wigman On the Robin spectrum for the equilateral triangle . J. Phys. A: Math. Theor. 55 254004 (2022). special issue ‘Claritons and Asymptotics of Ideas: The Physics of Michael Berry’. doi.org/10.1088/1751-8121/ac6f9b

  85. Zeev Rudnick and Niclas Technau. The metric theory of the pair correlation function for small non-integer powers. J. Lond. Math. Soc. (2) 106 (2022), no. 3, 2752--2772. DOI: 10.1112/jlms.12647

  86. Zeev Rudnick GOE statistics on the moduli space of surfaces of large genus . Geom. Funct. Anal. 33 (2023), no. 6, 1581–1607. doi.org/10.1007/s00039-023-00655-6

  87. Zeev Rudnick and Igor Wigman On the Central Limit Theorem for linear eigenvalue statistics on random surfaces of large genus . Journal d'Analyse Mathematique 151, 293–302 (2023). doi.org/10.1007/s11854-023-0327-7

  88. Zeev Rudnick. Zeros of modular forms and Faber polynomials. Mathematika, Volume 70, Issue 2 (2024). doi.org/10.1112/mtk.12244 arXiv:2308.08352 [math.NT]

Books

  1. A. Granville and Z. Rudnick (eds), Equidistribution in number theory, an introduction, Proceedings of NATO Advanced Study Institute on Equidistribution in Number Theory, Montreal 2005 Nato Science Series vol 237, Springer 2007.

Surveys and other articles

  1. Z. Rudnick Poincare series and Kloosterman sums over a function field. (Based on lectures by I. Piatetski-Shapiro) in ``The Arithmetic and Spectral Analysis of Poincare Series" by J. Cogdell and I. Piatetski-Shapiro, Perspectives in Mathematics 13, Academic Press 141-182 (1990).

  2. Z. Rudnick Zeta functions in arithmetic and their spectral statistics Proceedings of a special semester at the Institut Poincare, 1996 (unpublished).

  3. Z. Rudnick On quantum unique ergodicity for linear maps of the torus , in proceedings of the third European Congress of Mathematicians, July 2000, Barcelona.

  4. Z. Rudnick Number Theoretic Background Proceedings of a summer school in Bologna, August 2001.

  5. Z. Rudnick Eigenvalue statistics and lattice points Colloquio DeGeorgi, Pisa 2006.

  6. Z. Rudnick The arithmetic theory of quantum maps Proceedings of NATO Advanced Study Institute on Equidistribution in Number Theory, Montreal 2005, Nato Science Series vol 237, Springer 2007.

  7. A. Granville and Z. Rudnick Uniform distribution Proceedings of NATO Advanced Study Institute on Equidistribution in Number Theory, Montreal 2005, Nato Science Series vol 237, Springer 2007.

  8. A. Granville and Z. Rudnick Torsion points on curves Proceedings of NATO Advanced Study Institute on Equidistribution in Number Theory, Montreal 2005, Nato Science Series vol 237, Springer 2007.

  9. Z. Rudnick What is Quantum Chaos , Notices of the AMS, 55 number 1 (2008), 32--34.
    Chinese version , in Mathematical Advance in Translation, sponsored by the Academy of Mathematics and Systems Science (Chinese Academy of Sciences).

  10. Z. Rudnick Some problems in analytic number theory for polynomials over a finite field. Proceedings of the ICM vol 2 (2014), 443--460.

  11. Z. Rudnick Tribute to Jean Bourgain , at the memorial ceremony, IAS Princeton, May 2019. Appeared in "Remembering Jean Bourgain (1954–2018)", Notices of the AMS vol 68, number 6 (2021), DOI: https://doi.org/10.1090/noti2290

  12. Zeev Rudnick On the sum of digits of 1/M in Fq[x]. arXiv:2108.10142 [math.NT]

Ph.D. thesis (Yale, 1990).