## Counting zeros of Dedekind zeta functions

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Elchin Hasanalizade, Quanli Shen and Peng-Jie Wong
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**91**(2022), 277-293

## Abstract:

Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $\zeta _K(s)$ of $K$. More precisely, we show that for $T \geq 1$, \begin{equation*} \Big | N_K (T) - \frac {T}{\pi } \log \Big ( d_K \Big ( \frac {T}{2\pi e}\Big )^{n_K}\Big )\Big | \le 0.228 (\log d_K + n_K \log T) + 23.108 n_K + 4.520, \end{equation*} which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett et al. on counting zeros of Dirichlet $L$-functions.## References

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## Additional Information

**Elchin Hasanalizade**- Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada
- ORCID: 0000-0003-2604-7221
- Email: e.hasanalizade@uleth.ca
**Quanli Shen**- Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada
- MR Author ID: 1166842
- ORCID: 0000-0003-3095-3702
- Email: quanli.shen@uleth.ca
**Peng-Jie Wong**- Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada
- MR Author ID: 1211484
- Email: pengjie.wong@uleth.ca
- Received by editor(s): July 18, 2020
- Received by editor(s) in revised form: April 5, 2021
- Published electronically: July 19, 2021
- Additional Notes: The second and third authors are the corresponding authors. This research was supported by the NSERC Discovery grants RGPIN-2020-06731 of Habiba Kadiri and RGPIN-2020-06032 of Nathan Ng. The third author was supported by a PIMS postdoctoral fellowship and the University of Lethbridge
- © Copyright 2021 Elchin Hasanalizade, Quanli Shen, and Peng-Jie Wong
- Journal: Math. Comp.
**91**(2022), 277-293 - MSC (2020): Primary 11R42
- DOI: https://doi.org/10.1090/mcom/3665
- MathSciNet review: 4350540