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Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications
January 28, 2016 12:00–12:25, Dorodnitsyn Computing Centre, Department of Mechanics and Mathematics of Lomonosov Moscow State University., 119991, Moscow, Gubkina str., 8, Steklov Mathematical Institute, 9 floor, Conference hall

Karatsuba's method of estimating of short Kloosterman sums

M. A. Korolev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
 Video records: Flash Video 195.8 Mb Flash Video 1,167.3 Mb MP4 195.8 Mb

$$S(m,A)=\sum_{n\in A}\exp{(\frac{an^{*}+bn}{m})},\quad nn^*\equiv 1 \pmod m.$$
Here $m>2$, $A$ denotes some subset of reduced residual system $\mathbb{Z}_{m}^{*}$ modulo $m$, such that its cardinality $|A|$ does not exceed arbitrary small fixed power of $m$. In the talk, we will discuss the last results concerning such sums and the further development of Karatsuba's method.