## On functional properties of incomplete Gaussian sums.(English)Zbl 0728.11039

Incomplete Gaussian sums are exponential sums $$\sum_{n\in \omega}\exp (2\pi ian^ 2/q)$$, where $$(a,q)=1$$ and $$\omega$$ is an interval of length at most q. These can be expressed in terms of the function $H(x_ 1,x_ 2)=\sum_{n\neq 0}\frac{\exp (2\pi i(n^ 2x_ 2+nx_ 1))}{2\pi in},$ where the sum of the series is understood as the limit of the symmetric partial sums. By an analysis of this series, it is proved, for instance, that if $$\omega$$ runs over a system of nonoverlapping intervals in the interval [0,q], then for each positive $$\epsilon >0$$, the number of intervals such that the modulus of the corresponding incomplete Gaussian sums exceeds $$\epsilon\sqrt{q}$$ is at most $$c\epsilon^{-2}$$, where c is an absolute positive constant. Also, the size of the implied constant in the classical estimate $$\ll \sqrt{q}$$ for incomplete Gaussian sums is discussed.
Reviewer: M.Jutila (Turku)

### MSC:

 11L05 Gauss and Kloosterman sums; generalizations

### Keywords:

Incomplete Gaussian sums; number of intervals; constant
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