## Explicit bound for the number of primes in arithmetic progressions assuming the Generalized Riemann Hypothesis

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Anne-Maria Ernvall-Hytönen and Neea Palojärvi
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## Abstract:

We prove an explicit error term for the $\psi (x,\chi )$ function assuming the Generalized Riemann Hypothesis. Using this estimate, we prove a conditional explicit bound for the number of primes in arithmetic progressions.## References

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## Additional Information

**Anne-Maria Ernvall-Hytönen**- Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki, Finland
- Email: anne-maria.ernvall-hytonen@helsinki.fi
**Neea Palojärvi**- Affiliation: Matematik och Statistik, Åbo Akademi University, Domkyrkotorget 1, 20500 Åbo, Finland
- Address at time of publication: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki, Finland
- ORCID: 0000-0001-7749-8730
- Email: neea.palojarvi@helsinki.fi
- Received by editor(s): March 4, 2020
- Received by editor(s) in revised form: May 26, 2020, December 1, 2020, June 13, 2021, and August 11, 2021
- Published electronically: February 15, 2022
- Additional Notes: The work of the first author was supported by the Emil Aaltonen foundation.
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 1317-1365 - MSC (2020): Primary 11N13, 11Y35; Secondary 11B25, 11M26
- DOI: https://doi.org/10.1090/mcom/3691
- MathSciNet review: 4405497