New estimates for short Kloosterman sums with weights

2022 ◽  
Vol 86 (3) ◽  
Author(s):  
Natalia Kirillovna Semenova
Keyword(s):  
Kloosterman Sums

scholarly journals Short Kloosterman Sums for Polynomials over Finite Fields

2003 ◽  
Vol 55 (2) ◽  
pp. 225-246 ◽  
Author(s):  
William D. Banks ◽  
Asma Harcharras ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Small Degree ◽  
Kloosterman Sums ◽  
Degree Relative ◽  
Irreducible Polynomials ◽  
Arbitrary Polynomial ◽  
Polynomials Over Finite Fields ◽  
Titchmarsh Theorem

AbstractWe extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring [x]/M(x) for collections of polynomials either of the form f−1g−1 or of the form f−1g−1 + afg, where f and g are polynomials coprime to M and of very small degree relative to M, and a is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

scholarly journals The distribution of the maximum of partial sums of Kloosterman sums and other trace functions

2021 ◽  
Vol 157 (7) ◽  
pp. 1610-1651
Author(s):  
Pascal Autissier ◽  
Dante Bonolis ◽  
Youness Lamzouri
Keyword(s):  
Distribution Function ◽  
Recent Work ◽  
Character Sums ◽  
Kloosterman Sums ◽  
Partial Sums ◽  
Optimal Range ◽  
Uniform Estimates ◽  
The Third ◽  
Distribution Of The Maximum ◽  
Complex Valued

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$ -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$ -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.

scholarly journals An Upper Bound for the Extended Kloosterman Sums over Galois Rings

1998 ◽  
Vol 4 (3) ◽  
pp. 218-238 ◽  
Author(s):  
Abhijit G. Shanbhag ◽  
P. Vijay Kumar ◽  
Tor Helleseth
Keyword(s):  
Upper Bound ◽  
Kloosterman Sums ◽  
Galois Rings

scholarly journals Divisibility properties of classical binary Kloosterman sums

2009 ◽  
Vol 309 (12) ◽  
pp. 3975-3984 ◽  
Author(s):  
Pascale Charpin ◽  
Tor Helleseth ◽  
Victor Zinoviev
Keyword(s):  
Kloosterman Sums ◽  
Divisibility Properties

On a Problem of D. H. Lehmer and Kloosterman Sums

2003 ◽  
Vol 139 (3) ◽  
pp. 247-257 ◽  
Author(s):  
Zhang Wenpeng
Keyword(s):  
Kloosterman Sums
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