manuscripta mathematica

, Volume 118, Issue 2, pp 135–149

A proof of Selberg's orthogonality for automorphic L-functions


DOI: 10.1007/s00229-005-0563-4

Cite this article as:
Liu, J., Wang, Y. & Ye, Y. manuscripta math. (2005) 118: 135. doi:10.1007/s00229-005-0563-4


Let π and π′ be automorphic irreducible cuspidal representations of GLm(QA) and GLm(QA), respectively. Assume that π and π′ are unitary and at least one of them is self-contragredient. In this article we will give an unconditional proof of an orthogonality for π and π′, weighted by the von Mangoldt function Λ(n) and 1−n/x. We then remove the weighting factor 1−n/x and prove the Selberg orthogonality conjecture for automorphic L-functions L(s,π) and L(s,π′), unconditionally for m≤4 and m′≤4, and under the Hypothesis H of Rudnick and Sarnak [20] in other cases. This proof of Selberg's orthogonality removes such an assumption in the computation of superposition distribution of normalized nontrivial zeros of distinct automorphic L-functions by Liu and Ye [12].


Automorphic L-function Selberg's orthogonality 

Mathematics Subject Classification (2000)

11F70 11N05 11F66 11M26 11M41 

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsShandong UniversityJinanChina
  2. 2.Department of MathematicsCapital Normal UniversityBeijingChina
  3. 3.Department of MathematicsThe University of IowaIowa CityUSA