The circle method and bounds for 𝐿-functions—III: 𝑡-aspect subconvexity for 𝐺𝐿(3) 𝐿-functions

Abstract

Let π \pi be a Hecke-Maass cusp form for S L ( 3 , Z ) SL(3,\mathbb {Z}) . In this paper we will prove the following subconvex bound: \[ L ( 1 2 + i t , π ) ≪ π , ε ( 1 + | t | ) 3 4 − 1 16 + ε . L\left (\tfrac {1}{2}+it,\pi \right )\ll _{\pi ,\varepsilon } (1+|t|)^{\frac {3}{4}-\frac {1}{16}+\varepsilon }. \]