The shifted convolution L-function for Maass forms

• Published in: Research in number theory 2024-12, Vol.10 (4), Article 86
• Main Authors: Goldfeld, Dorian, Hinkle, Gerhardt, Hoffstein, Jeffrey
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics; Number Theory
• ISSN: 2522-0160; 2363-9555
• DOI: 10.1007/s40993-024-00575-w

Let Φ 1 , Φ 2 be Maass forms for S L ( 2 , Z ) with Fourier coefficients C 1 ( n ) , C 2 ( n ) . For a positive integer h the meromorphic continuation and growth in s ∈ C (away from poles) of the shifted convolution L-function L h ( s , Φ 1 , Φ 2 ) : = ∑ n ≠ 0 , - h C 1 ( n ) C 2 ( n + h ) · | n ( n + h ) | - 1 2 s is obtained. For Re ( s ) > 0 it is shown that the only poles are possible simple poles at 1 2 ± i r k , where 1 4 + r k 2 are eigenvalues of the Laplacian. As an application we obtain, for T → ∞ , the asymptotic formula ∑ | n ( n + h ) | < T n ≠ 0 , - h C 1 ( n ) C 2 ( n + h ) l o g ( T | n ( n + h ) | ) 3 2 + ε = f r 1 , r 2 , h , ε ( T ) · T 1 2 + O h 1 - ε T ε + h 1 + ε T - 2 - 2 ε , where the function f r 1 , r 2 , h , ε ( T ) is given as an explicit spectral sum that satisfies the bound f r 1 , r 2 , h , ε ( T ) ≪ h θ + ε . We also obtain a sharp bound for the above shifted convolution sum with sharp cutoff, i.e., without the smoothing weight log ( ∗ ) 3 2 + ε with uniformity in the h aspect. Specifically, we show that for h < x 1 2 - ε , ∑ | n ( n + h ) | < x C 1 ( n ) C 2 ( n + h ) ≪ h 2 3 θ + ε x 2 3 ( 1 + θ ) + ε + h 1 2 + ε x 1 2 + 2 θ + ε .