Generalized Kloosterman sum with primes

M. A. Korolev

Informatics and Automation, Analytic and combinatorial number theory, Tome 296 (2017), pp. 163-180.

Résumé

The work is devoted to generalized Kloosterman sums modulo a prime, i.e., trigonometric sums of the form $\sum _{p\le x}\exp \{2\pi i (a\overline {p}\,{+}\,F_k(p))/q\}$ and $\sum _{n\le x}\mu (n)\exp \{2\pi i (a\overline {n}\,{+}\,F_k(n))/q\}$, where $q$ is a prime number, $(a,q)=1$, $m\overline {m}\equiv 1\pmod q$, $F_k(u)$ is a polynomial of degree $k\ge 2$ with integer coefficients, and $p$ runs over prime numbers. An upper estimate with a power saving is obtained for the absolute values of such sums for $x\ge q^{1/2+\varepsilon }$.

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