Second Moment of the GL_3 Standard L-function on the Critical Line

The second and fourth moments of the Riemann zeta function have been known for about a century, but the sixth moment remains elusive.  

The sixth moment of zeta can be thought of as the second moment of a GL_3 Eisenstein series, and it is natural to consider variants of the problem

where the Eisenstein series is replaced by a cusp form.  I will discuss recent work with Agniva Dasgupta and Wing Hong Leung where we obtain a

nontrivial bound on this second moment.  I will also discuss some applications, including an improvement on the Rankin-Selberg problem.