Mollified Moments of Cubic Dirichlet L-Functions over the Eisenstein Field

We prove, assuming the generalized Riemann Hypothesis (GRH) that the density of the $L$-functions associated with primitive cubic Dirichlet characters over the Eisenstein field that do not vanish at the central point $s=1/2$ is positive. This is achieved by computing the first mollified moment assuming a subconvexity bound, and obtaining a sharp upper bound for the higher mollified moments for these $L$-functions under GRH. The proportion of non-vanishing is explicit, but extremely small.Comment: 46 page

Waring–Goldbach Problem with Piatetski-Shapiro Primes

In this paper, we exhibit an asymptotic formula for the number of representations of a large integer as a sum of a fixed power of Piatetski-Shapiro primes, thereby establishing a variant of Waring-Goldbach problem with primes from a sparse sequence