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AVERAGES OF EXPONENTIAL TWISTS OF THE VON MANGOLDT FUNCTION

Part of: Zeta and $L$-functions: analytic theory Exponential sums and character sums

Published online by Cambridge University Press:  25 April 2022

XIUMIN REN  and
WEI ZHANG Open the ORCID record for WEI ZHANG [Opens in a new window]
XIUMIN REN
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, PR China e-mail: [email protected]
WEI ZHANG*
Affiliation:
School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, PR China
*
e-mail: [email protected]
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Abstract

We obtain some improved results for the exponential sum $\sum _{x<n\leq 2x}\Lambda (n)e(\alpha k n^{\theta })$ with $\theta \in (0,5/12),$ where $\Lambda (n)$ is the von Mangoldt function. Such exponential sums have relations with the so-called quasi-Riemann hypothesis and were considered by Murty and Srinivas [‘On the uniform distribution of certain sequences’, Ramanujan J. 7 (2003), 185–192].

Keywords

exponential sums over primeszero-density estimates

MSC classification

Primary: 11L20: Sums over primes
Secondary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 3 , December 2022 , pp. 425 - 430
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work was supported by National Natural Science Foundation of China (Grant No. 11871307).

References

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