A prime number theorem in short intervals for dihedral Maass newforms

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.

Keywords

• prime number theorem
• hecke eigenvalue
• rankin-selberg $ l $-function
• zero-free region
• zero density estimate