Moments, bounds, and zero density for L-functions of mixed type cusp forms

Prime numbers have long been a source of interest in mathematical research. We know that there are infinitely many prime numbers, but their distribution is incredibly complicated. In this thesis, we look at a class of functions known as L-functions that encode information about prime numbers. More specifically, we look at the growth rate of these L-functions and related quantities. The estimates that we achieve on the growth rate of these L-functions are known as subconvexity bounds, which are more accurate than standard bounds. Each of these L-functions behaves interestingly on a region known as the critical strip. We know that these L-functions have many roots in this critical strip, and it is conjectured that all the roots lie on a line dividing the critical strip directly in half. While this task seems to be beyond the state of current mathematics, we are able to prove a statement about the growth rate of the number of zeros in this critical strip, providing corroborative evidence for related conjectures.