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 | References | Similar Articles | Additional Information

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