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On Sums Involving Quadratic Characters
Journal of the London Mathematical Society, 1967The authors consider the sum \(S(k)=\sum_{n=1}^{p-1}({n\over p})\) where \(p\equiv 3\pmod 4\) and \(({n\over p})\) is Legendre's symbol. Using the theorem of \textit{P. T. Bateman, S. Chowla} and \textit{P. Erdős} [Publ. Math. 1, 165--182 (1950; Zbl 0036.30702)] concerning the size of \(\sum_{n=1}^\infty\frac{({n\over p})}{n}\) the authors show that ...
Ayoub, R. G. +2 more
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Local and Global Character Sums
Acta Mathematica Sinica, English Series, 2000The author establishes a multiplicative analog (for character sums) of Hua's theorem on the estimate of complete trigonometric sums in an algebraic number field. On the other hand, a relationship between liftings of a character sum in a local field is also studied.
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Journal of the London Mathematical Society, 1971
asterisk (*) meansthat the singularities (modp) of r, and r2 are excluded and in the sum 1 lo (o f 0 (modp)) is to be interpreted as the unique integer w (modp) such that ow = 1 (modp)). Perel'muter has given conditions under which this sum is O(pf), thus generalizing the earlier deep work of Weil [4] and Carlitz and Uchiyama [2].
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asterisk (*) meansthat the singularities (modp) of r, and r2 are excluded and in the sum 1 lo (o f 0 (modp)) is to be interpreted as the unique integer w (modp) such that ow = 1 (modp)). Perel'muter has given conditions under which this sum is O(pf), thus generalizing the earlier deep work of Weil [4] and Carlitz and Uchiyama [2].
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Quaestiones Mathematicae, 2017
Let q ≥ 2 be an integer, X be a non-principal character mod q, A = A(q) ≤ q, B = B(q) = q, and H = H(q) ≤ q. In this paper, we shall use the estimates for Kloosterman sums and the properties of trigonometric sums to give some sharp estimates for character sums of the formTk(x, A, B,H; q) = ∑ ak X (a), a∈h(A ...
Ren, Ganglian +2 more
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Let q ≥ 2 be an integer, X be a non-principal character mod q, A = A(q) ≤ q, B = B(q) = q, and H = H(q) ≤ q. In this paper, we shall use the estimates for Kloosterman sums and the properties of trigonometric sums to give some sharp estimates for character sums of the formTk(x, A, B,H; q) = ∑ ak X (a), a∈h(A ...
Ren, Ganglian +2 more
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Bulletin of the London Mathematical Society, 1988
The author estimates the sum \[ A=\sum_ ...
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The author estimates the sum \[ A=\sum_ ...
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Jacobi Sums and Hecke Characters
American Journal of Mathematics, 1985Let K be an abelian number field contained in \({\mathbb{Q}}(\zeta_ N)\). \textit{A. Weil} [Trans. Am. Math. Soc. 73, 487-495 (1952; Zbl 0048.270); Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1974, 1-14 (1974; Zbl 0367.10035)] showed how to use Jacobi sums attached to characters of order dividing N to produce Hecke characters whose values lie in ...
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Mean values of character sums III
Mathematika, 1987The author continues his investigation on mean value estimates for character sums, started in [Mathematika 33, 1-5 (1986; Zbl 0595.10028) and ibid. 34, 1-7 (1987; Zbl 0626.10036)]. In the present paper he proves the following theorem: Let \(p\) be an odd prime number and \(h< p^3\) a positive integer. For any Dirichlet character \(\chi\) modulo \(p^3\)
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Trigonometric Sums and Characters
1982Let m be a positive integer. We have seen that the set of integers can be partitioned into residue classes $$A_0 ,A_1 , \ldots ,A_{m - 1}$$
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