Abstract:
A new elementary proof of an estimate for incomplete Kloosterman sums modulo a prime $q$ is obtained. Along with Bourgain's 2005 estimate of the double Kloosterman sum of a special form, it leads to an elementary derivation of an estimate for Kloosterman sums with primes for the case in which the length of the sum is of order $q^{0.5+\varepsilon}$, where $\varepsilon$ is an arbitrarily small fixed number.
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