Heilbronn-like sums and their properties

H. Saydi and M. R. Darafsheh
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 3, Pages 104—112
DOI: 10.7546/nntdm.2021.27.3.104-112
Download full paper: PDF, 227 Kb


Authors and affiliations

H. Saydi
College of Mathematical Science, Tarbiat Modares University
Tehran, Iran

M. R. Darafsheh
School of Mathematics, Statistics and Computer Science
College of Science, University of Tehran, Tehran, Iran


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).


  • Supercharacter
  • Heilbronn sum
  • Supercharacter table

2020 Mathematics Subject Classification

  • 20C15
  • 11T23


  1. André, C. A. M. (1995). Basic characters of the unitriangle group. Journal of Algebra, 175(1), 287–319.
  2. André, C. A. M. (2001). The basic character table of the unitriangular group. Journal of Algebra, 241(1), 437–471.
  3. André, C. A. M. (2002). Basic characters of the unitriangle group (for arbitrary prime). Proceedings of the American Mathematical Society, 130(7), 1943–1954.
  4. Brumbaugh, J. L., Bulkow, M., Fleming, P. S., German, L. A. G., Garcia, S. R., Karaali, G., Michal, M., Turner. A. P., & Suh, H. (2014). Supercharacters, exponential sums, and the uncertainty principle. Journal of Number Theory, 144, 151–175.
  5. Diaconis, P., & Isaacs, I. M. (2008). Supercharacters and superclasses for algebra groups. Transactions of the American Mathematical Society, 360(5), 2359–2392.
  6. Dornhoff, L. (1971). Group Representation Theory. Part A: Ordinary Representation Theory, Marcel Dekker, Inc., New York.
  7. Garcia, S. R., & Lutz, B. (2018). A supercharacter approach to Heilbronn sums. Journal of Number Theory, 186, 1–15.

Related papers

Cite this paper

Saydi, H., & Darafsheh, M. R. (2021). Heilbronn-like sums and their properties. Notes on Number Theory and Discrete Mathematics, 27(3), 104-112, doi: 10.7546/nntdm.2021.27.3.104-112.

Comments are closed.