August 1, 2025

Second Moment of Degree Three L-Functions

Key Points

An upper bound of the second moment of |L(F,1/2+it)|^2 improves existing results on subconvexity for self-dual l-functions.
The derived bound is T^{3/2-3/32+ε}, establishing a non-trivial error term related to self-dual gl(3) l-functions.
Approach involves evaluation of the integral of |L(F,1/2+it)|^2 across the t-aspect domain.
The findings highlight potential advancements in error estimates for the rankin-selberg problem and zero density for gl(3) l-functions.

Abstract

Abstract Let F be a Hecke–Maaß cusp form for SL (3, Z). We obtain the first non-trivial upper bound of the second moment of L (F, s) in t-aspect: align* & ₓ^2T|L (F, 1/2+it) |² dt₅, T^3/2-3/32+. align* Immediate corollaries include improvements over the existing results on the subconvexity bound for self-dual GL (3) L-functions in the t-aspect and for self-dual GL (3) GL (2) L-functions in the GL (2) spectral aspect, the error term in the Rankin–Selberg problem, and the zero density estimate for GL (3) L-functions.

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Second Moment of Degree Three L-Functions

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Abstract

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