Abstract
For the completed Riemann zeta function
Acknowledgements
The authors thank the referee for valuable comments and corrections which have improved the quality of this paper. The first author would also like to thank the Mathematics Department at Rice University for their hospitality where part of this research was completed.
References
[1] A. Akbary and M. R. Murty, Uniform distribution of zeros of Dirichlet series, Anatomy of Integers, CRM Proc. Lecture Notes 46, American Mathematical Society, Providence (2008), 143–158. 10.1090/crmp/046/10Search in Google Scholar
[2] H. M. Bui, Gaps between zeros of the derivative of the Riemann ξ-function, J. Théor. Nombres Bordeaux 22 (2010), no. 2, 287–305. 10.5802/jtnb.716Search in Google Scholar
[3]
M. W. Coffey,
Asymptotic estimation of
[4] B. Conrey, Zeros of derivatives of Riemann’s ξ-function on the critical line, J. Number Theory 16 (1983), no. 1, 49–74. 10.1016/0022-314X(83)90031-8Search in Google Scholar
[5] B. Conrey and H. Iwaniec, Spacing of zeros of Hecke L-functions and the class number problem, Acta Arith. 103 (2002), no. 3, 259–312. 10.4064/aa103-3-5Search in Google Scholar
[6] J. B. Conrey, 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. 10.1515/crll.1989.399.1Search in Google Scholar
[7] T. Craven, G. Csordas and W. Smith, The zeros of derivatives of entire functions and the Pólya–Wiman conjecture, Ann. of Math. (2) 125 (1987), no. 2, 405–431. 10.2307/1971315Search in Google Scholar
[8] H. Davenport and H. Montgomery, Multiplicative Number Theory, Grad. Texts Math. 74, Springer, New York, 2013. Search in Google Scholar
[9] P. Erdös, On the integers having exactly K prime factors, Ann. of Math. (2) 49 (1948), 53–66. 10.2307/1969113Search in Google Scholar
[10] P. Erdös and P. Turán, On a problem in the theory of uniform distribution. I, Nederl. Akad. Wetensch., Proc. 51 (1948), 1146–1154; Indagationes Math. 10, 370–378 (1948). Search in Google Scholar
[11] D. W. Farmer, S. M. Gonek and Y. Lee, Pair correlation of the zeros of the derivative of the Riemann ξ-function, J. Lond. Math. Soc. (2) 90 (2014), no. 1, 241–269. 10.1112/jlms/jdu026Search in Google Scholar
[12] D. W. Farmer and R. C. Rhoades, Differentiation evens out zero spacings, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3789–3811. 10.1090/S0002-9947-05-03721-9Search in Google Scholar
[13] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University, Cambridge, 2009. 10.1017/CBO9780511801655Search in Google Scholar
[14] K. Ford, K. Soundararajan and A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function. II, Math. Ann. 343 (2009), no. 3, 487–505. 10.1007/s00208-008-0280-xSearch in Google Scholar
[15] K. Ford and A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, J. Reine Angew. Math. 579 (2005), 145–158. 10.1515/crll.2005.2005.579.145Search in Google Scholar
[16] A. Fujii, On a theorem of Landau, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 2, 51–54. 10.3792/pjaa.65.51Search in Google Scholar
[17] A. Fujii, Some problems of Diophantine approximation in the theory of the Riemann zeta function. III, Comment. Math. Univ. St. Pauli 42 (1993), no. 2, 161–187. Search in Google Scholar
[18] S. M. Gonek, An explicit formula of Landau and its applications to the theory of the zeta-function, A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemp. Math. 143, American Mathematical Society, Providence (1993), 395–413. 10.1090/conm/143/1210528Search in Google Scholar
[19] E. Hlawka, Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafunktion zusammenhängen, Österreich. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 184 (1975), no. 8–10, 459–471. 10.1007/978-3-642-35384-0_30Search in Google Scholar
[20] A. Ivić, The Riemann Zeta-function. The Theory of the Riemann Zeta-function with Applications, John Wiley & Sons, New York, 1985. Search in Google Scholar
[21] H. Ki, The Riemann Ξ-function under repeated differentiation, J. Number Theory 120 (2006), no. 1, 120–131. 10.1016/j.jnt.2005.11.004Search in Google Scholar
[22] H. Ki and Y.-O. Kim, On the number of nonreal zeros of real entire functions and the Fourier–Pólya conjecture, Duke Math. J. 104 (2000), no. 1, 45–73. 10.1215/S0012-7094-00-10413-9Search in Google Scholar
[23] Y.-O. Kim, Critical points of real entire functions and a conjecture of Pólya, Proc. Amer. Math. Soc. 124 (1996), no. 3, 819–830. 10.1090/S0002-9939-96-03083-3Search in Google Scholar
[24] E. Landau, Über die Nullstellen der Zetafunktion, Math. Ann. 71 (1912), no. 4, 548–564. 10.1007/BF01456808Search in Google Scholar
[25] J. E. Littlewood, On the Riemann zeta-function, Proc. Lond. Math. Soc. (2) 24 (1925), no. 3, 175–201. 10.1112/plms/s2-24.1.175Search in Google Scholar
[26] H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic Number Theory (St. Louis 1972), Proc. Sympos. Pure Math. 24, American Mathematical Society, Providence (1973), 181–193. 10.1090/pspum/024/9944Search in Google Scholar
[27] H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, J. Lond. Math. Soc. (2) 8 (1974), 73–82. 10.1112/jlms/s2-8.1.73Search in Google Scholar
[28] M. R. Murty and A. Perelli, The pair correlation of zeros of functions in the Selberg class, Int. Math. Res. Not. IMRN 1999 (1999), no. 10, 531–545. 10.1155/S1073792899000276Search in Google Scholar
[29] R. Murty and A. Zaharescu, Explicit formulas for the pair correlation of zeros of functions in the Selberg class, Forum Math. 14 (2002), no. 1, 65–83. 10.1515/form.2002.006Search in Google Scholar
[30] A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273–308. 10.1090/S0025-5718-1987-0866115-0Search in Google Scholar
[31] H. Rademacher, Topics in Analytic Number Theory, Grundlehren Math. Wiss. 169, Springer, New York, 1973. 10.1007/978-3-642-80615-5Search in Google Scholar
[32] H. Rademacher, Collected Papers of Hans Rademacher. Vol. II, MIT Press, Cambridge, 1974. Search in Google Scholar
[33] A. Selberg, On the zeros of Riemann’s zeta-function, Skr. Norske Vid. Akad. Oslo I. 10 (1942), 1–59. Search in Google Scholar
[34] E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., The Clarendon Press, New York, 1986. Search in Google Scholar
[35] H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352. 10.1007/BF01475864Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On the distribution of zeros of derivatives of the Riemann ξ-function
- Representations of constant socle rank for the Kronecker algebra
- Abstract bivariant Cuntz semigroups II
- Curves on Segre threefolds
- 𝒩(p, q, s)-type spaces in the unit ball of ℂn(II): Carleson measure and its application
- On a class of critical Robin problems
- On the description of multidimensional normal Hausdorff operators on Lebesgue spaces
- Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures
- Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits
- Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation
- Some remarks on products of sets in the Heisenberg group and in the affine group
- Codimension growth of central polynomials of Lie algebras
- Unramified Whittaker functions for certain Brylinski–Deligne covering groups
- Tilting classes over commutative rings
Articles in the same Issue
- Frontmatter
- On the distribution of zeros of derivatives of the Riemann ξ-function
- Representations of constant socle rank for the Kronecker algebra
- Abstract bivariant Cuntz semigroups II
- Curves on Segre threefolds
- 𝒩(p, q, s)-type spaces in the unit ball of ℂn(II): Carleson measure and its application
- On a class of critical Robin problems
- On the description of multidimensional normal Hausdorff operators on Lebesgue spaces
- Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures
- Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits
- Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation
- Some remarks on products of sets in the Heisenberg group and in the affine group
- Codimension growth of central polynomials of Lie algebras
- Unramified Whittaker functions for certain Brylinski–Deligne covering groups
- Tilting classes over commutative rings