共查询到20条相似文献,搜索用时 0 毫秒
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Christoph Aistleitner 《Mathematische Zeitschrift》2013,275(3-4):681-688
Bourgain posed the problem of calculating $$\begin{aligned} \Sigma = \sup _{n \ge 1} ~\sup _{k_1 < \cdots < k_n} \frac{1}{\sqrt{n}} \left\| \sum _{j=1}^n e^{2 \pi i k_j \theta } \right\| _{L^1([0,1])}. \end{aligned}$$ It is clear that $\Sigma \le 1$ ; beyond that, determining whether $\Sigma < 1$ or $\Sigma =1$ would have some interesting implications, for example concerning the problem whether all rank one transformations have singular maximal spectral type. In the present paper we prove $\Sigma \ge \sqrt{\pi }/2 \approx 0.886$ , by this means improving a result of Karatsuba. For the proof we use a quantitative two-dimensional version of the central limit theorem for lacunary trigonometric series, which in its original form is due to Salem and Zygmund. 相似文献
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REN Xiumin Department of Mathematics Shandong University Jinan China 《中国科学A辑(英文版)》2005,48(6):785-797
In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2 log κ/log2, x≥2 and α=a/q λsubject to (a, q) = 1, 1≤a≤q, and λ∈R. Then As an application, we prove that with at most O(N2/8 ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis. 相似文献
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Ondrej uch 《Finite Fields and Their Applications》2005,11(4):700-723
In this paper we compute geometric monodromy groups of additive exponential sums over BbbAn. Our approach builds on work of N. Katz, and involves p-adic analysis of explicit sums and computation of the Galois group of an equation over a function field in characteristic 2. The paper also provides a brief historical outline of the problem and lists previously known results. 相似文献
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Let denote the finite field of order q of characteristic p. We study the p-adic valuations for zeros of L-functions associated with exponential sums of the following family of Laurent polynomials where , . When , the estimate of the associated exponential sum appears in Iwaniecʼs work on small eigenvalues of the Laplace–Beltrami operator acting on automorphic functions with respect to the group , and Adolphson and Sperber gave complex absolute values for zeros of the corresponding L-function. Using the decomposition theory of Wan, we determine the generic Newton polygon (q-adic values of the reciprocal zeros) of the L-function. Working on the chain level version of Dworkʼs trace formula and using Wanʼs decomposition theory, we are able to give an explicit Hasse polynomial for the generic Newton polygon in low dimensions, i.e., . 相似文献
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M. Jutila 《Proceedings Mathematical Sciences》1987,97(1-3):157-166
Let τ(n) be the arithmetical function of Ramanujan, α any real number, and x≥2. The uniform estimate $$\mathop \Sigma \limits_{n \leqslant x} \tau (n)e(n\alpha ) \ll x^6 \log x$$ is a classical result of J R Wilton. It is well known that the best possible bound would be ?x 6. The validity of this hypothesis is proved. 相似文献
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H. Maier 《Journal of Number Theory》2009,129(7):1669-1677
Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity
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Prof. Alex Bacopoulos 《Numerische Mathematik》1970,16(3):243-247
A theory of sums of Chebyshev approximations is useful for the problem of simultaneous minimization of the absolute and relative errors of an approximation. In this paper some of the important properties of the Chebyshev alternation theory are studied from the point of view of extending them to sums of Chebyshev norms. Both positive and negative results are obtained. Specifically, it is shown that the sum of Chebyshev approximations with different weight functions is not a Chebyshev approximation. 相似文献
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Steven I Sperber 《Journal of Number Theory》1980,12(2):141-153
By use of p-adic analytic methods, we study the L-functions associated to certain exponential sums defined over a finite field. Estimates for the degree of this L-function as rational function are obtained. In an “asymptotic” sense, these estimates are shown to be best possible. Precise determination of the degree is computed in the one-variable case. 相似文献
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In this paper, we use the elementary and analytic methods to study the computational problem of one kind mean value involving
the classical Dedekind sums and two-term exponential sums, and give two exact computational formulae for them. 相似文献
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On a problem concerning spacings 总被引:1,自引:0,他引:1
Shihong Cheng 《Probability Theory and Related Fields》1984,66(2):245-258
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Bing He 《The Ramanujan Journal》2017,43(2):313-326
For any integer \(n> 1,\) we prove The first three results confirm three divisibility properties on sums of binomial coefficients conjectured by Z.-W. Sun.
相似文献
$$\begin{aligned} 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(3k+1){2k\atopwithdelims ()k}^3(-8)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(6k+1){2k\atopwithdelims ()k}^3(-512)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(42k+5){2k\atopwithdelims ()k}^3 4096^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(20k^2+8k+1){2k\atopwithdelims ()k}^5(-4096)^{n-1-k}. \end{aligned}$$
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Let G be a connected graph with v(G) 2 vertices and independence number (G). G is critical if for any edge e of G:
1. (i) (G − e) > (G), if e is not a cut edge of G, and
2. (ii) v(Gi) − (Gi) < v(G) − (G), I = 1, 2, if e is a cut edge and G1, G2 are the two components of G − e.
Recently, Katchalski et al. (1995) conjectured that: if G is a connected critical graph, then with equality possible if and only if G is a tree. In this paper we establish this conjecture. 相似文献
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A. S. Besicovitch 《Mathematische Annalen》1938,115(1):613-618