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Mathematics of Computation

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Counting zeros of Dirichlet $L$-functions


Authors: Michael A. Bennett, Greg Martin, Kevin O’Bryant and Andrew Rechnitzer
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
Published electronically: January 26, 2021
MathSciNet review: 4232231
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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 $\sigma <-1/2$.


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Additional 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.
Article copyright: © Copyright 2021 by the authors