scholarly journals On the identity involving certain Hardy sums and Kloosterman sums

2014 ◽  
Vol 2014 (1) ◽  
Author(s):  
Han Zhang ◽  
Wenpeng Zhang
Keyword(s):  
Kloosterman Sums ◽  
Certain Hardy Sums

scholarly journals Some Identities Involving Certain Hardy Sums and General Kloosterman Sums

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 95 ◽  
Author(s):  
Huifang Zhang ◽  
Tianping Zhang
Keyword(s):  
Mean Value ◽  
Gauss Sums ◽  
Kloosterman Sums ◽  
Orthogonality Relation ◽  
Character Sum ◽  
Hybrid Mean Value ◽  
Principal Character ◽  
Modulo P ◽  
Certain Hardy Sums ◽  
The Mean

Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums K ( r , l , λ ; p ) and certain Hardy sums S 1 ( h , q ) ∑ m = 1 p − 1 ∑ s = 1 p − 1 K ( m , n , λ ; p ) K ( s , t , λ ; p ) S 1 ( 2 m s ¯ , p ) , ∑ m = 1 p − 1 ∑ s = 1 p − 1 | K ( m , n , λ ; p ) | 2 | K ( s , t , λ ; p ) | 2 S 1 ( 2 m s ¯ , p ) . Our results not only cover the previous results, but also contain something quite new. Actually the previous authors just consider the case of the principal character λ modulo p, while we consider all the cases.

Estimates for Kloosterman sums for totally real number fields

2001 ◽  
Vol 2001 (535) ◽  
Author(s):  
R. W. Bruggeman ◽  
R. J. Miatello ◽  
I. Pacharoni
Keyword(s):  
Real Number ◽  
Number Fields ◽  
Kloosterman Sums ◽  
Totally Real

ON THE GENERAL k-TH KLOOSTERMAN SUMS AND ITS FOURTH POWER MEAN

2004 ◽  
Vol 25 (01) ◽  
pp. 97-102 ◽  
Author(s):  
HONGYAN LIU ◽  
WENPENG ZHANG
Keyword(s):  
Kloosterman Sums ◽  
Fourth Power ◽  
Power Mean ◽  
Fourth Power Mean

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.

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