@article{21385,
  abstract     = {We prove that the average size of a mixed character sum (math. formular) (for a suitable smooth function w) is on the order of √x for all irrational real θ satisfying a weak Diophantine condition, where χ is drawn from the family of Dirichlet characters modulo a large prime r and where x 6 r. In contrast, it was proved by Harper that the average size is o(√x) for rational θ. Certain quadratic Diophantine equations play a key role in the present paper. },
  author       = {Wang, Victor and Xu, Max},
  issn         = {1473-7124},
  journal      = {Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
  pages        = {1--15},
  publisher    = {Cambridge University Press},
  title        = {{Average sizes of mixed character sums}},
  doi          = {10.1017/prm.2026.10123},
  year         = {2026},
}