Equidistribution of Eisenstein series on geodesic segments Academic Article

abstract

• 2018 Elsevier Inc. We show an asymptotic formula for the L2 norm of the Eisenstein series E(z,1/2+iT) restricted to a segment of a geodesic connecting infinity and an arbitrary real. For generic geodesics of this form, the asymptotic formula shows that the Eisenstein series satisfies restricted QUE. On the other hand, for rational geodesics, the Eisenstein series does not satisfy restricted quantum ergodicity. As an application, we show that the zero set of the Eisenstein series intersects every such geodesic segment, provided T is large. We also make analogous conjectures for the Maass cusp forms. In particular, we predict that cusp forms do not satisfy restricted quantum ergodicity for rational geodesics.

authors

• Young, Matthew

published proceedings

• ADVANCES IN MATHEMATICS

altmetric score

• 1.25

author list (cited authors)

• Young, M. P.

citation count

• 2

complete list of authors

• Young, Matthew P

publication date

• December 2018

publisher

• Elsevier  Publisher

published in

• Advances in Mathematics  Journal

keywords

• Dirichlet L-functions
• Eisenstein Series
• Equidistribution
• Quantum Ergodicity
• Riemann Zeta Function

Digital Object Identifier (DOI)

• 10.1016/j.aim.2018.10.030

start page

• 1166

end page

• 1218

volume

• 340

URL

• http://dx.doi.org/10.1016/j.aim.2018.10.030