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

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- 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