Special values of symmetric power L-functions and Hecke eigenvalues

On a conjecture of Montgomery-Vaughan on extreme values of automorphic 𝐿-functions at 1

“…The values of these functions at the edge of the critical strip contain information of great interest. For example, Serre [18] showed that the Sato-Tate conjecture is equivalent to L(1 + iτ, sym m f ) = 0 for all m ∈ N and τ ∈ R. The distribution of the values L(1, sym m f ) has received attention of many authors, including Goldfeld, Hoffstein & Lieman [2], Hoffstein & Lockhart [7], Luo [12], Royer [14,15], Royer & Wu [16,17], Cogdell & Michel [1], Habsieger & Royer [5] and Lau & Wu [10,11]. In particular, Lau & Wu ([10], [11]) proved the following results:…”

Distribution of values of L -functions at the edge of the critical strip

“…The distribution of these values have been extensively studied over the last decades. One can quote the work of Granville-Soundararajan [7] and Lamzouri [15] in the case of ζ(1 + it); Elliott ( [5] and [6]), Montgomery-Vaughan [20] and Granville-Soundararajan [8] in the case of Dirichlet L-functions of quadratic characters L(1, χ d ); Duke [4] in the case of Artin L-functions, and the work of Cogdell-Michel [1], Habsieger-Royer [9], Lau-Wu [16], Liu-Royer-Wu [19], Royer ([21] and [22]), and Royer-Wu ( [23] and [24]) in the case of symmetric power L-functions of GL(2)-automorphic forms.…”

A Mean Value Result for a Product of Gl(2) and Gl(3) ‐functions

“…Then we have the following asymptotic formula. This is an example of mixed moments that were previously investigated, for example, in [16,19,20,24].…”

“…1 ≪ 1 for all n ≥ 0. Then, for any fixed A > 0, we have This is an example of mixed moments that were previously investigated, for example, in [16,19,20,24].…”