The shifted convolution L-function for Maass forms

2023/11/11 by Goldfeld, Dorian, Hinkle, Gerhardt, Hoffstein, Jeffrey
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.06587

Abstract

Let Φ₁,Φ₂ be Maass forms for SL(2,\mathbb Z) with Fourier coefficients C₁(n),C₂(n). For a positive integer h the meromorphic continuation and growth in s∈\mathbb C (away from poles) of the shifted convolution L-function L_h(s,Φ₁,Φ₂) := ∑_{n ≠ 0,-h} C₁(n) C₂(n + h) ⋅ |n(n + h)|^-(1)/(2)s is obtained. For \rm Re(s) > 0 it is shown that the only poles are possible simple poles at (1)/(2) ± ir_k, where \tfrac14+r_k² are eigenvalues of the Laplacian. As an application we obtain, for T→∞, the asymptotic formula \beginalign* & \undersetn ≠ 0,-h∑_√(|n (n + h)|)

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