On the average trigonometric sums for solutions of quadratic comparisons

In 1963, Hawley [1] (see also [2]), based on estimates of average trigonometric sums from solutions of quadratic comparisons, proved for the first time an asymptotic formula for the average number of divisors of a quadratic polynomial with a power-law decrease in the residual term. These results were later reinforced in [3] and [4]. In the present paper, we prove stronger results on this topic.