Bounding the Order of Vanishing of Cuspidal Newforms Via the Nth Centered Moments

Building on the work of Iwaniec, Luo and Sarnak, we use the n-level density to bound the probability of vanishing to order at least r at the central point for families of cuspidal newforms of prime level N →∞ , split by sign. There are three methods to improve bounds on the order of vanishing: optimizing the test functions, increasing the support, and increasing the n-level density studied. Previous work has determined the optimal test functions for the 1 and 2-level densities in certain support ranges, with the effectiveness of the bounds only marginally increasing by the optimized test functions over simpler ones, and thus this is not expected to be a productive avenue for further research. Similarly the support has been increased as far as possible, and further progress is shown to be related to delicate and difficult conjectures in number theory. Thus we concentrate on the third method, and study the higher centered moments (which are similar to the n-level densities but combinatorially easier). We find the level at each rank for which the upper bounds on the order of vanishing is the best, thus producing world-record bounds on the order of vanishing to rank at least r for every r > 2 (for example, our bounds for vanishing to order at least 5 or at least 6 are less than half the previous bounds, a significant improvement). Additionally, we calculate the bound using the optimal test function for the 1-level density from previous work and compare it to the naive test functions for higher levels. We find that the optimal test function for certain levels are not the optimal for other levels, and some test functions may outperform others for some levels but not in others. Finally, we calculate the integrals needed to determine the bounds, doing so by transforming an n-dimensional integral to a 1-dimensional integral and greatly reducing the computation cost in the process.