Estimates of Weyl’s exponential sums for very small values of the variable

Abstract.

We consider Weyl’s exponential sums for very small values of the variable. In this case we can give an asymptotic transformation formula. Weyl’s exponential sums will be usually estimated by means of Weyl’s or Vinogradov’s method. Here we use van der Corput’s method and obtain sufficiently good results in the present case.