Abstract
Let π and π′ be automorphic irreducible cuspidal representations of GL m (Q A ) and GL m ′(Q A ), 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].
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Liu, J., Wang, Y. & Ye, Y. A proof of Selberg's orthogonality for automorphic L-functions. manuscripta math. 118, 135–149 (2005). https://doi.org/10.1007/s00229-005-0563-4
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DOI: https://doi.org/10.1007/s00229-005-0563-4