Counting zeros of Dirichlet $L$-functions
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- by Michael A. Bennett, Greg Martin, Kevin O’Bryant and Andrew Rechnitzer;
- Math. Comp. 90 (2021), 1455-1482
- DOI: https://doi.org/10.1090/mcom/3599
- Published electronically: January 26, 2021
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Abstract:
We give explicit upper and lower bounds for $N(T,\chi )$, the number of zeros of a Dirichlet $L$-function with character $\chi$ and height at most $T$. Suppose that $\chi$ has conductor $q>1$, and that $T\geq 5/7$. If $\ell =\log \frac {q(T+2)}{2\pi }> 1.567$, then \begin{equation*} \left | N(T,\chi ) - \left ( \frac {T}{\pi } \log \frac {qT}{2\pi e} -\frac {\chi (-1)}{4}\right ) \right | \le 0.22737 \ell + 2 \log (1+\ell ) - 0.5. \end{equation*} We give slightly stronger results for small $q$ and $T$. Along the way, we prove a new bound on $|L(s,\chi )|$ for $\sigma <-1/2$.References
- G. Alirezaei, A Sharp Double Inequality for the Inverse Tangent Function, available at arXiv:1307.4983.
- Michael A. Bennett, Greg Martin, Kevin O’Bryant, and Andrew Rechnitzer, Explicit bounds for primes in arithmetic progressions, Illinois J. Math. 62 (2018), no. 1-4, 427–532. MR 3922423, DOI 10.1215/ijm/1552442669
- Richard P. Brent, On the accuracy of asymptotic approximations to the log-gamma and Riemann-Siegel theta functions, J. Aust. Math. Soc. 107 (2019), no. 3, 319–337. MR 4034593, DOI 10.1017/s1446788718000393
- H. M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 466039
- D. E. G. Hare, Computing the principal branch of log-Gamma, J. Algorithms 25 (1997), no. 2, 221–236. MR 1478568, DOI 10.1006/jagm.1997.0881
- Kevin S. McCurley, Explicit estimates for the error term in the prime number theorem for arithmetic progressions, Math. Comp. 42 (1984), no. 165, 265–285. MR 726004, DOI 10.1090/S0025-5718-1984-0726004-6
- The PARI Group, PARI/GP version 2.11.0, Univ. Bordeaux (2018), available at http://pari.math.u-bordeaux.fr/.
- David J. Platt, Numerical computations concerning the GRH, Math. Comp. 85 (2016), no. 302, 3009–3027. MR 3522979, DOI 10.1090/mcom/3077
- Hans Rademacher, On the Phragmén-Lindelöf theorem and some applications, Math. Z. 72 (1959/60), 192–204. MR 117200, DOI 10.1007/BF01162949
- Atle Selberg, Contributions to the theory of Dirichlet’s $L$-functions, Skr. Norske Vid.-Akad. Oslo I 1946 (1946), no. 3, 62. MR 22872
- Flemming Topsøe, Some bounds for the logarithmic function, Inequality theory and applications. Vol. 4, Nova Sci. Publ., New York, 2007, pp. 137–151. MR 2349596
- Timothy S. Trudgian, An improved upper bound for the argument of the Riemann zeta-function on the critical line II, J. Number Theory 134 (2014), 280–292. MR 3111568, DOI 10.1016/j.jnt.2013.07.017
- T. S. Trudgian, An improved upper bound for the error in the zero-counting formulae for Dirichlet $L$-functions and Dedekind zeta-functions, Math. Comp. 84 (2015), no. 293, 1439–1450. MR 3315515, DOI 10.1090/S0025-5718-2014-02898-6
Bibliographic Information
- Michael A. Bennett
- Affiliation: Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2 Canada
- MR Author ID: 339361
- Email: bennett@math.ubc.ca
- Greg Martin
- Affiliation: Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2 Canada
- MR Author ID: 619056
- ORCID: 0000-0002-8476-9495
- Email: gerg@math.ubc.ca
- Kevin O’Bryant
- Affiliation: Department of Mathematics, City University of New York, College of Staten Island and The Graduate Center, 2800 Victory Boulevard, Staten Island, New York 10314
- MR Author ID: 667411
- Email: kevin.obryant@csi.cuny.edu
- Andrew Rechnitzer
- Affiliation: Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, V6T 1Z2 Canada
- MR Author ID: 626723
- ORCID: 0000-0002-4386-3207
- Email: andrewr@math.ubc.ca
- Received by editor(s): May 5, 2020
- Received by editor(s) in revised form: August 25, 2020
- Published electronically: January 26, 2021
- Additional Notes: The first, second, and fourth authors were supported by NSERC Discovery Grants. Support for this project was provided to the third author by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.
- © Copyright 2021 by the authors
- Journal: Math. Comp. 90 (2021), 1455-1482
- MSC (2020): Primary 11N13, 11N37, 11M20, 11M26; Secondary 11Y35, 11Y40
- DOI: https://doi.org/10.1090/mcom/3599
- MathSciNet review: 4232231