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Canadian Mathematical Bulletin, 1973
Let p be an odd prime, n an integer not divisible by p and α a positive integer. For any integer h with (h,pα)=l, is defined as any solution of the congruence (mod,pα). The Kloosterman sum Ap α(n) (see for example [4]) is defined by(1.1)where the dash (') indicates that the letter of summation runs only through a reduced residue system with respect to
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Let p be an odd prime, n an integer not divisible by p and α a positive integer. For any integer h with (h,pα)=l, is defined as any solution of the congruence (mod,pα). The Kloosterman sum Ap α(n) (see for example [4]) is defined by(1.1)where the dash (') indicates that the letter of summation runs only through a reduced residue system with respect to
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On the General Kloosterman Sums
Journal of Mathematical Sciences, 2005zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On the Values of Kloosterman Sums
IEEE Transactions on Information Theory, 2009Given a prime p and a positive integer n, we show that the shifted Kloosterman sums SigmaxisinF p nPsi(x + alphaxpn-2)=SigmaxisinF* p nPsi(x+alphax-1)+1, alphaisinF*pn where Psi is a nontrivial additive character of a finite field Fpn of pn elements, do not vanish if alpha belongs to a small subfield Fpm sube Fpn.
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2020
These are a set of notes that introduces the classical Kloosterman sums and proves their basic properties. Kloosterman's bound is proved for the sums which is weaker than the sharp Weil bound. Bounds due to Esterman are also proved and also Selberg's identity is proved for Kloosterman sums.
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These are a set of notes that introduces the classical Kloosterman sums and proves their basic properties. Kloosterman's bound is proved for the sums which is weaker than the sharp Weil bound. Bounds due to Esterman are also proved and also Selberg's identity is proved for Kloosterman sums.
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Mathematika, 1961
Let m, n, q denote positive integers, p a prime, and a, b, h, r, s, t, u, v integers.
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Let m, n, q denote positive integers, p a prime, and a, b, h, r, s, t, u, v integers.
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Kloosterman Sums and their Applications: A Review
Results in Mathematics, 1996This paper lists results, and explains some concepts and results, in the areas of automorphic forms (holomorphic as well as real analytic ones) for cofinite discrete groups of motions in the upper half plane, Kloosterman sums, Hecke operators, Selberg trace formula, Kuznetsov sum formula, representation of \(SL_2\) over the adeles.
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Short Kloosterman Sums with Primes
Mathematical Notes, 2019zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On Kloosterman Sums with Oscillating Coefficients
Canadian Mathematical Bulletin, 1999AbstractIn this paper we prove: for any positive integers a and q with (a, q) = 1, we have uniformlyThis improves the previous bound obtained by D. Hajela, A. Pollington and B. Smith [5].
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A note on the moments of Kloosterman sums
Applicable Algebra in Engineering, Communication and Computing, 2009This paper considers several types of Kloosterman sums and proves identities between such sums. Three main types are considered: Kloosterman sums \(K_n(a)\) of degree \(n\), \(m\)-dimensional Kloosterman sums \(K^{(m)}(a)\), and \(m\)-dimensional Kloosterman sums \(K_n^{(m)}(a)\) of degree \(n\). In order to define these, we recall some notation: \(p\)
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On a generalization of Kloosterman sums
Mathematical Notes, 2015For integers \(u, v, w\) and natural numbers \(q\), \(d\) with \(d\mid q\). We define \[ K_{q,d}\left(u, v; w\right)=\mathop{\mathop{\sum_{z=1}^{q}}_{(z,q)=1}}_{z\equiv w \;(\bmod d)} e\left(\frac{uz+vz^{-1}}{q}\right), \] where \(e(x)=\text{e}^{2\pi ix}\).
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