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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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Doklady Mathematics, 2022
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Russian Academy of Sciences. Izvestiya Mathematics, 1993
In the paper under review the classical Kloosterman sum \[ K(d_ 1, d_ 2;Q) = \sum_{{x_ 1, x_ 2 \text{mod} Q \atop x_ 1x_ 2 \equiv 1 \pmod Q}} \exp \left( 2\pi i {d_ 1x_ 2 + d_ 2 x_ 2 \over Q} \right) \] is expressed in terms of numbers connected with the arithmetic of \(\mathbb{Z}/m \mathbb{Z}\) where ...
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In the paper under review the classical Kloosterman sum \[ K(d_ 1, d_ 2;Q) = \sum_{{x_ 1, x_ 2 \text{mod} Q \atop x_ 1x_ 2 \equiv 1 \pmod Q}} \exp \left( 2\pi i {d_ 1x_ 2 + d_ 2 x_ 2 \over Q} \right) \] is expressed in terms of numbers connected with the arithmetic of \(\mathbb{Z}/m \mathbb{Z}\) where ...
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Journal of Mathematical Sciences, 2005
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Izvestiya: Mathematics, 1995
Kloosterman sums are of the type \[ S(a,b;m)=\sum _{1\leq n\leq m, (m,n)=1} e((an^*+bn)/m), \] where \(nn^*\equiv 1\pmod{m}\). The author restricts the sum here to integers \(n=xy\) with \((xy,m)=1\) and \(x\), \(y\) lying in certain intervals. A complicated but perfectly explicit bound is given for such a modified sum. The bilinear shape of the sum is
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Kloosterman sums are of the type \[ S(a,b;m)=\sum _{1\leq n\leq m, (m,n)=1} e((an^*+bn)/m), \] where \(nn^*\equiv 1\pmod{m}\). The author restricts the sum here to integers \(n=xy\) with \((xy,m)=1\) and \(x\), \(y\) lying in certain intervals. A complicated but perfectly explicit bound is given for such a modified sum. The bilinear shape of the sum is
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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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