## Riesz-type criteria for the Riemann hypothesis

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- by Archit Agarwal, Meghali Garg and Bibekananda Maji PDF
- Proc. Amer. Math. Soc.
**150**(2022), 5151-5163 Request permission

## Abstract:

In 1916, Riesz proved that the Riemann Hypothesis is equivalent to the bound $\sum _{n=1}^\infty \frac {\mu (n)}{n^2} \exp \left ( - \frac {x}{n^2} \right ) = O_{\epsilon } \left ( x^{-\frac {3}{4} + \epsilon } \right )$, as $x \rightarrow \infty$, for any $\epsilon >0$. Around the same time, Hardy and Littlewood gave another equivalent criterion for the Riemann Hypothesis while correcting an identity of Ramanujan. In the present paper, we establish a one-variable generalization of the identity of Hardy and Littlewood and as an application, we provide Riesz-type criteria for the Riemann Hypothesis. In particular, we obtain the bound given by Riesz as well as the bound of Hardy and Littlewood.## References

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## Additional Information

**Archit Agarwal**- Affiliation: Department of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, Madhya Pradesh 453552, India
- ORCID: 0000-0003-2046-2758
- Email: archit.agrw@gmail.com, phd2001241002@iiti.ac.in
**Meghali Garg**- Affiliation: Department of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, Madhya Pradesh 453552, India
- ORCID: 0000-0001-6748-4618
- Email: meghaligarg.2216@gmail.com, phd2001241005@iiti.ac.in
**Bibekananda Maji**- Affiliation: Department of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, Madhya Pradesh 453552, India
- MR Author ID: 1181160
- ORCID: 0000-0003-2155-2480
- Email: bibek10iitb@gmail.com, bibekanandamaji@iiti.ac.in
- Received by editor(s): July 1, 2021
- Received by editor(s) in revised form: February 23, 2022
- Published electronically: August 18, 2022
- Additional Notes: The third author was supported by Start-up Research Grant, SERB # SRG/2020/00144
- Communicated by: Benjamin Brubaker
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**150**(2022), 5151-5163 - MSC (2020): Primary 11M06; Secondary 11M26
- DOI: https://doi.org/10.1090/proc/16064