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Kloosterman sums in residue rings. (English) Zbl 1319.11051

The authors in this paper study the properties of inversion modulo \(m\) where \(m\) is not a prime number. The authors denote the inverse of \(x\) modulo \(m\) by \(x^{*}\) so their main focus is to study the properties of the following set: \[ I(N) = \{x^{*}\bmod m\;|\; \gcd(x,m)=1\; \text{and}\;x\in\{1,\ldots, N\}\}, \] where \(\{1,\ldots, N\}\) represents the integer numbers in the range \(1\) to \(N\). The first theorem gives a non trivial bound for small values of \(N\) for the number of solutions of the following equation, \[ x_1+\ldots + x_k = x_{k+1}+\ldots + x_{2k}\bmod m, \quad x_1,\ldots, x_{2k}\in I(N). \] The authors also provide bounds when the values \(x_1,\ldots, x_{2k}\) are primes and belong to \(I(N)\). The proofs are nice and use lattice techniques, which could be applied to other problems.
The other two results consists on bounds for exponential sums with general moduli, which are of the following form, \[ \sum_{x_1\in I(N_1)} \sum_{x_2\in I(N_2)}\alpha(x_1)\alpha(x_2)e_m(x_1^{*}x_2^{*}), \quad\text{and}\quad \sum_{x_1\in I(N)}e_m(a x_1^{*}), \] where \(e_m(x)\) is \(e^{2\pi x/m}\), \(\gcd(a,m)=1\) and \(\alpha\) is an application \(\alpha(x)\in\mathbb{C}\) with \(|\alpha(x)|\leq 1\).

MSC:

11L05 Gauss and Kloosterman sums; generalizations
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