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# On a New Exponential Sum

Let $p$ be prime and let $\vartheta\in\Z^*_p$ be of multiplicative order $t$ modulo $p$. We consider exponential sums of the form $$S(a) = \sum_{x =1}^{t} \exp(2\pi i a \vartheta^{x^2}/p)$$ and prove that for any $\varepsilon > 0$ $$\max_{\gcd(a,p) = 1} |S(a)| = O( t^{5/6 + \varepsilon}p^{1/8}) .$$
 MSC Classifications: 11L07 - Estimates on exponential sums 11T23 - Exponential sums 11B50 - Sequences (mod $m$) 11K31 - Special sequences 11K38 - Irregularities of distribution, discrepancy [See also 11Nxx]