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Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications
January 28, 2016 12:00, 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
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M. A. Korolev


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Abstract: In the middle of 1990th, A.A. Karatsuba invented a new and original method of estimating of short Kloosterman sums, that is, the exponential sums of the folloving type:
$$ S(m,A)=\sum_{n\in A}\exp{(\frac{an^{*}+bn}{m})},\quad nn^*\equiv 1 \pmod m. $$
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.

Language: Russian and English

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