Number of Nonzero Cubic Sums

Аннотация

The exponential sums \( {S}_q\left(a,m\right)=\sum \limits_{l=1}^q\exp \left(2\pi i\left({al}^3+ ml\right){q}^{-1}\right) \) are considered. For every positive integer q, closed-form expressions for the number of nonzero sums occurring among S_q(a, 0), . . . , S_q(a, q − 1) are found.