Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on simple $ a$-points of $ L$-functions

Authors: S. M. Gonek, S. J. Lester and M. B. Milinovich
Journal: Proc. Amer. Math. Soc. 140 (2012), 4097-4103
MSC (2010): Primary 11M06, 11M26
Published electronically: April 10, 2012
MathSciNet review: 2957199
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Abstract: We prove, subject to certain hypotheses, that a positive proportion of the $ a$-points of the Riemann zeta-function and Dirichlet $ L$-functions with primitive characters are simple and discuss corresponding results for other functions in the Selberg class. We also prove an unconditional result of this type for the $ a$-points in fixed strips to the right of the line $ \Re s=1/2$.

References [Enhancements On Off] (What's this?)

  • 1. Peter J. Bauer, Zeros of Dirichlet 𝐿-series on the critical line, Acta Arith. 93 (2000), no. 1, 37–52. MR 1760087
  • 2. Bruce C. Berndt, The number of zeros for 𝜁^{(𝑘)}(𝑠), J. London Math. Soc. (2) 2 (1970), 577–580. MR 0266874
  • 3. Vibeke Borchsenius and Børge Jessen, Mean motions and values of the Riemann zeta function, Acta Math. 80 (1948), 97–166. MR 0027796
  • 4. H. M. Bui, J. B. Conrey, and M. P. Young, More than 41% of the zeros of the zeta function are on the critical line, to appear in Acta Arith. Available on the arXiv at
  • 5. R. Garunkštis and J. Steuding, On the roots of the equation $ \zeta (s)=a$, preprint. Available on the arXiv at
  • 6. Norman Levinson, Almost all roots of 𝜁(𝑠)=𝑎 are arbitrarily close to 𝜎=1/2, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 1322–1324. MR 0406952
  • 7. Norman Levinson and Hugh L. Montgomery, Zeros of the derivatives of the Riemann zetafunction, Acta Math. 133 (1974), 49–65. MR 0417074
  • 8. K. Powell, Topics in analytic number theory, master's thesis, Brigham Young University, Provo, Utah, 2009.
  • 9. Atle Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno, Salerno, 1992, pp. 367–385. MR 1220477
  • 10. A. Speiser, Geometrisches zur Riemannschen Zeta Funktion, Math. Ann. 110 (1934), 514-521.
  • 11. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
  • 12. Kai-Man Tsang, THE DISTRIBUTION OF THE VALUES OF THE RIEMANN ZETA-FUNCTION, ProQuest LLC, Ann Arbor, MI, 1984. Thesis (Ph.D.)–Princeton University. MR 2633927
  • 13. S. M. Voronin, A theorem on the “universality” of the Riemann zeta-function, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 3, 475–486, 703 (Russian). MR 0472727

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Additional Information

S. M. Gonek
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

S. J. Lester
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

M. B. Milinovich
Affiliation: Department of Mathematics, University of Mississippi, University, Mississippi 38677

Keywords: $a$-points, simple zeros, Riemann zeta-function, $L$-functions, Selberg class
Received by editor(s): March 18, 2011
Received by editor(s) in revised form: May 18, 2011, and May 24, 2011
Published electronically: April 10, 2012
Additional Notes: Research of the first author was partially supported by NSF grant DMS-0653809.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.