AIM logo Brian Conrey

Brian Conrey

Executive Director
American Institute of Mathematics
600 E. Brokaw Road
San Jose, CA 95112
conrey@aimath.org
CV

Research Interests:

Analytic Number theory, especially the analytic theory of L-functions


Publications

  1. (Thesis) ZEROES OF DERIVATIVES OF RIEMANN'S XIFUNCTION ON THE CRITICAL LINE. Thesis (Ph.D.)–University of Michigan. 1980. 147 pp, ProQuest LLC.
  2. (with L.A. Rubel) On the location of the zeros of the derivatives of a polynomial, Proc. A.M.S., 86 (1982), 37--41.
  3. Zeros of derivatives of Riemann's xi-function on the critical line, J. Number Th., 16 (1983), 49--74.
  4. Zeros of derivatives of Riemann's xi-function on the critical line, J. Number Th., 17 (1983), 71--75.
  5. (with A. Ghosh and S.M. Gonek) A note on gaps between zeros of the zeta-function, Bull. London Math. Soc., 16 (1984), 421--424.
  6. (with A. Ghosh) On mean values of the zeta-function, Mathematika, 31 (1984), 159--161.
  7. (with A. Ghosh) A simpler proof of Levinson's theorem, Math. Proc. Cambridge Phil. Soc., 97 (1985), 385--395.
  8. (with A. Ghosh, D. Goldston, S.M. Gonek and D.R. Heath-Brown) Distribution of gaps between zeros of the zeta-function, Quarterly J. of Math. Oxford (2), 36 (1985), 43--51.
  9. (with R. Balasubramanian, and D.R. Heath-Brown) Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial, J. Reine Agnew. Math., 357 (1985), 161--181.
  10. On the distribution of the zeros of the Riemann zeta-function, in Topics in Analytic Number Theory, University of Texas Press, Austin, Texas, 1985.
  11. (with A. Ghosh) A mean value theorem for the Riemann zeta-function at its relative extrema on the critical line, J. London Math. Soc. (2) 32 (1985) 193--202.
  12. (with A. Ghosh and S.M. Gonek) Simple zeros of the zeta-function of a quadratic number field, I, Inventiones Math., 86 (1986), 536--576.
  13. (with A. Ghosh and S.M. Gonek) Large gaps between zeros of the zeta-functions, Mathematika, 33 (1986), 212--238.
  14. (with A. Ghosh and S.M. Gonek) Simple zeros of the zeta-function of a quadratic number field, II, in Analytic Number Theory and Diophantine Problems, Birkh\"{a}user, Boston $\cdot$ Basel $\cdot$ Stuttgart, 1987.
  15. The fourth moment of derivatives of the Riemann zeta-function, Quarterly J. Math. Oxford, (2) 39 (1988), 21--36.
  16. (with A. Ghosh) Simple zeros of the Dirichlet series formed with Ramanuja\'{n} tau--function, C. R. Math. Rep. Acad. Sci. Canada, 10, (1988), 95--99.
  17. (with L.A. Rubel) Erratum to "On the Location of the Zeros of the Derivative of a Polynomial", Proc. A.M.S., Vol. 104, No. 3. (Nov., 1988), p. 1002.
  18. (with A. Ghosh) On the zeros of the Taylor polynomials associated with the exponential function, American Math. Monthly, 95 (1988), 528--533.
  19. (with A. Ghosh and S. M. Gonek) Simple zeros of zeta functions. Colloque de Th\'{e}orie Analytique des Nombres "Jean Coquet" (Marseille, 1985), 77--83, Publ. Math. Orsay, 88-02, Univ. Paris XI, Orsay, 1988.
  20. (with A. Ghosh) Simple zeros of the Ramanujan $\tau$-Dirichlet series, Inventiones Math., 94 (1988), 403--419.
  21. At least two-fifths of the zeros of the Riemann zeta-function are on the critical line, Bull. AMS., 20 (1989), 79-81.
  22. (with Michael W. Smiley) Some geometric aspects of hyperbolic boundary value problems, Proc. American Math. Soc., 107 (1989), 591-601.
  23. (with A. Ghosh and S.M. Gonek) Mean values of the Riemannzeta-function with application to the distribution of zeros. Number theory, trace formulasand discrete groups (Oslo, 1987), 185–199, Academic Press, Boston, MA, 1989.
  24. More than two-fifths of the zeros of the Riemann zeta-function are on the critical line, J. Reine Angew. Math., 399 (1989), 1-26.
  25. (with A. Ghosh) On mean values of the zeta-function, II, Acta Arithmetica, LII (1989), 367-371.
  26. (with A. Ghosh) Zeros of derivatives of the Riemann zeta-function near the critical line. Analytic number theory (Allerton Park, IL, 1989), 95–110, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.
  27. (with A. Ghosh) Mean values of the Riemann zeta-function.III. Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), 35–59, Univ. Salerno, Salerno, 1992.
  28. (with A. Ghosh) On the Selberg class of Dirichlet series:small degrees. Duke Math. J. 72 (1993), no. 3, 673–693.
  29. (with A. Ghosh) Turan inequalities and zeros of Dirichlet series associated with certain Dirichlet series. Trans. American Math. Soc., 342 (1994), 407-419.
  30. (with D. Farmer) An extension of Hecke's converse theorem, International Mathematics Research Notices, 9 (1995) 445-463.
  31. (with E. Fransen, R. Klein, C. Scott) Meanvalues of Dedekind sums. J. Number Theory 56 (1996), no. 2, 214–226.
  32. A note on the fourth power moment of the Riemannzeta-function. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995), 225–230, Progr.Math., 138, Birkhäuser Boston, Boston, MA, 1996.
  33. (with W. Duke and D. Farmer) The distribution of the eigenvalues of Hecke operators, Acta Arithmetica, LXXVIII.4 (1997), 405-409.
  34. (with A. Ghosh and S. M. Gonek) Simple zeros of the Riemannzeta-function. Proc. London Math. Soc. (3) 76 (1998), no. 3, 497–522.
  35. (with A. Ghosh) A conjecture for the sixth power moment of the Riemann zeta-function. Internat. Math. Res. Notices 1998, no. 15, 775–780.
  36. (with D. W. Farmer) Hecke operators and the nonvanishing of $L$-functions. Topics in number theory (University Park, PA, 1997), 143–150, Math. Appl.,467, Kluwer Acad. ubl., Dordrecht, 1999.
  37. (with D. W. Farmer and K. Soundararajan) Transition meanvalues of real characters. J. Number Theory 82 (2000), no. 1, 109–120.
  38. (with H. Iwaniec) The cubic moment of central values of automorphic $L$-functions, Annals of Math. (2) 151 (2000), no. 3, 1175--1216.
  39. (with A. Granville, B. Poonen, and K. Soundararajan) Zeros of Fekete polynomials, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 3, 865--889.
  40. (with D. W. Farmer), Mean values of $L$-functions and symmetry. Internat. Math. Res. Notices 2000, no. 17, 883--908.
  41. (with Xian-Jin Li) A note on some positivity conditions related to zeta and $L$-functions, Internat. Math. Res. Notices 2000, no. 18, 929--940.
  42. (with D. W. Farmer and P. J. Wallace) Factoring Hecke polynomials modulo a prime, Pacific J. Math. 196 (2000), no. 1, 123--130.
  43. (with S. Gonek), High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), no. 3, 577--604.
  44. (with Xian-Jin Li) On the trace of Hecke operators for Maass forms for congruence subgroups, Forum Math. 13 (2001), no. 4, 447--484.
  45. $L$-functions and random matrices, Mathematics unlimited---2001 and beyond, 331--352, Springer, Berlin, 2001.
  46. (with H. Iwaniec) Spacing of zeros of Hecke $L$-functions and the class number problem, Acta Arith. 103 (2002), no. 3, 259--312.
  47. (with K. Soundararajan) Real zeros of quadratic Dirichlet L-functions, Invent. Math. 150 (2002), no. 1, 1--44.
  48. (with J. P. Keating, M. O. Rubinstein, and N. C. Snaith) On the frequency of vanishing of quadratic twists of modular $L$-functions, Number theory for the millennium, I (Urbana, IL, 2000), 301--315, A K Peters, Natick, MA, 2002.
  49. The Riemann hypothesis, Notices Amer. Math. Soc. 50 (2003), no. 3, 341--353.
  50. (with D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith) Autocorrelation of random matrix polynomials. Comm. Math. Phys. 237 (2003), no. 3, 365--395.
  51. (with P. J. Forrester and N. C. Snaith), Averages of ratios of characteristic polynomials for the compact classical groups. Int. Math. Res. Not. 2005, no. 7, 397--431.
  52. (with D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith), Integral Moments of L-functions, PLMS (3) 91 (2005) 33--104.
  53. Notes on eigenvalue distributions for the classical compact groups. Recent perspectives in random matrix theory and number theory, 111--145, London Math. Soc. Lecture Note Ser., 322, Cambridge Univ. Press, Cambridge, 2005.
  54. Families of $L$-functions and 1-level densities. Recent perspectives in random matrix theory and number theory, 225--249, London Math. Soc. Lecture Note Ser., 322, Cambridge Univ. Press, Cambridge, 2005.
  55. (with A. Gamburd) Pseudomoments of the Riemann zeta-function and pseudomagic squares. J. Number Theory 117 (2006), no. 2, 263--278.
  56. (with J. P. Keating, M. O. Rubinstein, and N. C. Snaith) Random matrix theory and the Fourier coefficients of half-integral-weight forms. Experiment. Math. 15 (2006), no. 1, 67--82.
  57. (with M. O. Rubinstein and N. C. Snaith) Moments of the derivative of characteristic polynomials with an application to the Riemann zeta function. Comm. Math. Phys. 267 (2006), no. 3, 611--629.
  58. Notes on L-functions and random matrix theory. Frontiers in number theory, physics, and geometry. I, 107--162, Springer, Berlin, 2006.
  59. (with A.\ Ghosh) Remarks on the generalized Lindelöf hypothesis. Funct. Approx. Comment. Math. 36 (2006), 71--78.
  60. (with D. Farmer, B. Odgers, and N. C. Snaith) A converse theorem for Γ0 (13), J. Number Theory, J. Number Theory 122 (2007), no. 2, 314--323.
  61. (with M.\ O.\ Rubinstein, N.\ C.\ Snaith,and M.\ Watkins) Discretisation for odd quadratic twists. Ranks of elliptic curves and random matrix theory, 201--214, London Math. Soc. Lecture Note Ser., 341, Cambridge Univ. Press, Cambridge, 2007.
  62. (with A.\ Pokharel, M.\ O.\ Rubinstein, and M.\ Watkins) Mark Secondary terms in the number of vanishings of quadratic twists of elliptic curve $L$-functions. Ranks of elliptic curves and random matrix theory, 215--232, London Math. Soc. Lecture Note Ser., 341, Cambridge Univ. Press, Cambridge, 2007.
  63. (with N. C. Snaith) Applications of the $L$-functions ratios conjecture, Proc. Lond. Math. Soc. (3) 94 (2007), no. 3, 594--646.
  64. (with D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith) Lower order terms in the full moment conjecture for the Riemann zeta function. J. Number Theory 128 (2008), no. 6, 1516--1554.
  65. (with N. C. Snaith) Triple correlation of the Riemann zeros. J. Théor. Nombres Bordeaux 20 (2008), no. 1, 61--106.
  66. (with N. C. Snaith) Correlations of eigenvalues and Riemann zeros. Commun. Number Theory Phys. 2 (2008), no. 3, 477--536.
  67. (with D. W. Farmer and Martin R. Zirnbauer) Autocorrelation of ratios of L-functions. Commun. Number Theory Phys. 2 (2008), no. 3, 593--636.
  68. (with D. W. Farmer and \Özlem Imamoglu) Palindromic random trigonometric polynomials. Proc. Amer. Math. Soc. 137 (2009), no. 5, 1835--1839.
  69. (with H. M. Bui and M. P. Young) More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (2011), no. 1, 35--64.
  70. Applications of the asymptotic large sieve. Actes de la Conférence ``Théorie des Nombres et Applications', 37--46, Publ. Math. Besançon Algèbre Théorie Nr., Presses Univ. Franche-Comté, Besançon, 2011.
  71. (with H. Iwaniec and K. Soundararajan) The sixth power moment of Dirichlet L-functions. Geom. Funct. Anal. 22 (2012), no. 5, 1257--1288.
  72. (with H. Iwaniec and K. Soundararajan) Small gaps between zeros of twisted L-functions. Acta Arith. 155 (2012), no. 4, 353--371.
  73. (with N. C. Snaith) On the orthogonal symmetry of L-functions of a family of Hecke Grössencharacters. Acta Arith. 157 (2013), no. 4, 323--356.
  74. (with S. Bettin) Period functions and cotangent sums. Algebra Number Theory 7 (2013), no. 1, 215--242.
  75. (with M. A. Holmstrom and T. L. McLaughlin) Smooth neighbors. Exp. Math. 22 (2013), no. 2, 195--202.
  76. (with D. W. Farmer and Özlem Imamoglu) The nontrivial zeros of period polynomials of modular forms Lie on the unit circle. Int. Math. Res. Not. IMRN 2013, no. 20, 4758--4771.
  77. (with S. Bettin) A reciprocity formula for a cotangent sum. Int. Math. Res. Not. IMRN 2013, no. 24, 5709--5726.
  78. (with H. Iwaniec and K. Soundararajan) Critical zeros of Dirichlet L-functions. J. Reine Angew. Math. 681 (2013), 175--198.
  79. (S. Bettin) The second weighted moment of ζ Actes de la Conférence "Théorie des Nombres et Applications'', 41--47, Publ. Math. Math. Besançon Algèbre Théorie Nr., 2013, Presses Univ. Franche-Comté, Besançon, 2013.
  80. (with S. Bettin and D.W. Farmer) An optimal choice of Dirichlet polynomials for the Nyman-Beurling criterion. Proc. Steklov Inst. Math. 280 (2013), suppl. 2, S30--S36.
  81. (with H. Iwaniec) Small populations of zeros of L-functions. Number theory, analysis, and combinatorics, 11--23, De Gruyter Proc. Math., De Gruyter, Berlin, 2014.
  82. (with N. C. Snaith) In support of $n$-correlation. Comm. Math. Phys. 330 (2014), no. 2, 639--653.
  83. Riemann's hypothesis. Colloquium De Giorgi 2013 and 2014, 109--117, Colloquia, 5, Ed. Norm., Pisa, 2014.
  84. (with J. Bober, D. W. Farmer, Akio Fujii, Sally Koutsoliotas, Stefan Lemurell, M. O. Rubinstein, and Hiroyuki Yoshida) The highest lowest zero of general L-functions. J. Number Theory 147 (2015), 364--373.
  85. (with J. P. Keating) Moments of zeta and correlations of divisor-sums: I. Philos. Trans. Roy. Soc. A 373 (2015), no. 2040, 20140313, 11 pp.
  86. (with J. P. Keating) Moments of zeta and correlations of divisor-sums: II. Advances in the theory of numbers, 75--85, Fields Inst. Commun., 77, Fields Inst. Res. Math. Sci., Toronto, ON, 2015
  87. (with J. P. Keating) Moments of zeta and correlations of divisor-sums: III. Indag. Math. (N.S.) 26 (2015), no. 5, 736--747.
  88. Lectures on the Riemann zeta function [book review of MR3241276]. Bull. Amer. Math. Soc. (N.S.) 53 (2016), no. 3, 507--512
  89. (with J. Gabbard, K. Grant, A. Liu, K. Morrison) Intransitive dice. Math. Mag. 89 (2016), no. 2, 133--143.
  90. Riemann's hypothesis. The legacy of Bernhard Riemann after one hundred and fifty years. Vol. I, 107--190, Adv. Lect. Math. (ALM), 35.1, Int. Press, Somerville, MA, 2016.
  91. (with J. P. Keating) Moments of zeta and correlations of divisor-sums: IV. Res. Number Theory 2 (2016), Art. 24, 24 pp.
  92. (with J. P. Keating) Pair correlation and twin primes revisited. Proc. A. 472 (2016), no. 2194, 20160548, 11 pp.
  93. (with J. P. Keating) Averages of ratios of the Riemann zeta-function and correlations of divisor sums. Nonlinearity 30 (2017), no. 10, R67--R80.
  94. (with C. Turnage-Butterbaugh) On r-gaps between zeros of the Riemann zeta-function. Bull. Lond. Math. Soc. 50 (2018), no. 2, 349--356.
  95. (with S. Bettin, J. Bober, A. Booker, M. Lee, G. Molteni, T. Oliver, D.Platt, and R. Steiner) A conjectural extension of Hecke's converse theorem. Ramanujan J. 47 (2018), no. 3, 659--684.
  96. (with J. P. Keating) Moments of zeta and correlations of divisor-sums: V. Proc. Lond. Math. Soc. (3) 118 (2019), no. 4, 729--752.
  97. (with E. C. Bailey, S. Bettin, G. Blower, A. Prokhorov, M. O.Rubinstein, N. C. Snaith) Mixed moments of characteristic polynomials of random unitary matrices. J. Math. Phys. 60 (2019), no. 8, 083509, 26 pp.
  98. (with Henryk Iwaniec, Kannan Soundararajan) The mean square of the product of a Dirichlet L-function and a Dirichlet polynomial. Funct. Approx. Comment. Math. 61 (2019), no. 2, 147–177.
  99. (with H. Iwaniec) Critical zeros of lacunary L-functions. ActaArith. 195 (2020), no. 3, 217–268.

Other published papers

  1. (with Paras M. Agrawal, Dan C. Sorescu, Ronald D. Kay, Donald L. Thompson, Lionel M. Raff, and A. Keith Jameson) Spectral line shapes in systems undergoing continuous frequency modulation. J. Chem. Phys. 105, 2686 (1996).
  2. Prime Time for the Riemann Hypothesis. Science 302 (2003), 60–61.
  3. The Riemann Hypothesis. Expeditions in Mathematics. (2011), 109–125.
  4. (with Mark Baker, Jane Beltran, Jason Buell, Tom Davis, Brianna Donaldson, Jeanne Detorre-Ozeki, Leila Dibble, Tom Freeman, Robert Hammie, Julie Montgomery, Avery Pickford, and Justine Wong) Sets, Planets, and Comets. College math. j. 44 (2013), no. 4, 258–264.

Unpublished papers

  1. On the Real and Imaginary Curves of the Riemann Zeta-Function.
  2. (with Huizhu Lu) A New Transform to Replace Fermat Transform for Digital Signal Processing.
  3. (with G. Myerson) On the Balazard-Saias Criterion for the Riemann Hypothesis. arXiv:math/0002254v1 [math.NT].
  4. Mollifying the Riemann Zeta-Function.
  5. Character sums and the Riemann Hypothesis.
  6. The Mean-Square of Dirichlet L-Functions. arXiv:0708.2699v1 [math.NT].
  7. RMT for Number Theorists.
  8. On the Length of the Continued Fraction of √q.
  9. (with Ben Simons and Igor Smolyarenko) Pair Correlation of Different L-Functions.
  10. (with David Farmer & Martin R. Zirnbauer) Autocorrelation of Ratios of Characteristic Polynomials and of L-FUnctions. 2004.
  11. (with David Farmer & M. R. Zirnbauer) Howe Pairs, Supersymmetry, and Ratios of Random Characteristic Polynomials for rhe Unitary Groups Un. arXiv:math-ph/0511024v2.
  12. An Example to Keep in Mind when Studying L-Functions.
  13. Moments of Z'Z: Some Calculations.
  14. Notes on Mollifiers.
  15. (with H. Iwaniec and K. Soundararajan) Asymptotic Large Sieve. arXiv:1105.1176v1 [math.NT].
  16. (with Jon Keating) Averages of Long Dirichlet Polynomials.
  17. (with Estelle Basor and Kent E. Morrison) Knots and Ones. arXiv:1703.00990v1 [math.GT] .

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