A note on simple $a$-points of $L$-functions
HTML articles powered by AMS MathViewer
- by S. M. Gonek, S. J. Lester and M. B. Milinovich PDF
- Proc. Amer. Math. Soc. 140 (2012), 4097-4103 Request permission
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
- Peter J. Bauer, Zeros of Dirichlet $L$-series on the critical line, Acta Arith. 93 (2000), no. 1, 37–52. MR 1760087, DOI 10.4064/aa-93-1-37-52
- Bruce C. Berndt, The number of zeros for $\zeta ^{(k)}\,(s)$, J. London Math. Soc. (2) 2 (1970), 577–580. MR 266874, DOI 10.1112/jlms/2.Part_{4}.577
- Vibeke Borchsenius and Børge Jessen, Mean motions and values of the Riemann zeta function, Acta Math. 80 (1948), 97–166. MR 27796, DOI 10.1007/BF02393647
- 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 http://arxiv.org/pdf/1002.4127v2.
- R. Garunkštis and J. Steuding, On the roots of the equation $\zeta (s)=a$, preprint. Available on the arXiv at http://arxiv.org/abs/1011.5339v1.
- Norman Levinson, Almost all roots of $\zeta (s)=a$ are arbitrarily close to $\sigma =1/2$, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 1322–1324. MR 406952, DOI 10.1073/pnas.72.4.1322
- Norman Levinson and Hugh L. Montgomery, Zeros of the derivatives of the Riemann zeta-function, Acta Math. 133 (1974), 49–65. MR 417074, DOI 10.1007/BF02392141
- K. Powell, Topics in analytic number theory, master’s thesis, Brigham Young University, Provo, Utah, 2009.
- 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
- A. Speiser, Geometrisches zur Riemannschen Zeta Funktion, Math. Ann. 110 (1934), 514–521.
- 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
- 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
- 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
Additional Information
- S. M. Gonek
- Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
- MR Author ID: 198665
- Email: gonek@math.rochester.edu
- S. J. Lester
- Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
- Email: lester@math.rochester.edu
- M. B. Milinovich
- Affiliation: Department of Mathematics, University of Mississippi, University, Mississippi 38677
- Email: mbmilino@olemiss.edu
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 4097-4103
- MSC (2010): Primary 11M06, 11M26
- DOI: https://doi.org/10.1090/S0002-9939-2012-11275-4
- MathSciNet review: 2957199