Heilbronn-like sums and their properties

Abstract

Heilbronn sums is of the form H_p(a)=\underset{l=1}{\overset{p-1}{\sum}}e(\dfrac{al^p}{p^2}), where p is an odd prime, and e(x)=\exp(2\pi ix). This is a supercharacter and has application in number theory. We extend this sum by defining D_p(a)=\underset{l=1}{\overset{p-1}{\sum}}e(\dfrac{al^p}{p^3}), where p is an odd prime and prove that D_p(a) is a supercharacter and drive a few identities involving D_p(a).