A prime number theorem in short intervals for dihedral Maass newforms

Bin Guan; Bin Guan

doi:10.3934/math.2024238

AIMS Mathematics

2024, Volume 9, Issue 2: 4896-4906. doi: 10.3934/math.2024238

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A prime number theorem in short intervals for dihedral Maass newforms

Bin Guan ^,

Data Science Institute, Shandong University, No. 27 Shanda South Road, Jinan 250100, China

Received: 17 November 2023 Revised: 12 January 2024 Accepted: 18 January 2024 Published: 23 January 2024
MSC : 11F30, 11N05

In this paper, we prove a prime number theorem in short intervals for the Rankin-Selberg $ L $-function $ L(s, \phi\times\phi) $, where $ \phi $ is a fixed dihedral Maass newform. As an application, we give a lower bound for the proportion of primes in a short interval at which the Hecke eigenvalues of the dihedral form are greater than a given constant.
- prime number theorem,
- Hecke eigenvalue,
- Rankin-Selberg $ L $-function,
- zero-free region,
- zero density estimate
Citation: Bin Guan. A prime number theorem in short intervals for dihedral Maass newforms[J]. AIMS Mathematics, 2024, 9(2): 4896-4906. doi: 10.3934/math.2024238

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Abstract

In this paper, we prove a prime number theorem in short intervals for the Rankin-Selberg $ L $-function $ L(s, \phi\times\phi) $, where $ \phi $ is a fixed dihedral Maass newform. As an application, we give a lower bound for the proportion of primes in a short interval at which the Hecke eigenvalues of the dihedral form are greater than a given constant.

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