Zero-sums of length kq in
Silke Kubertin (2005)
Acta Arithmetica
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Silke Kubertin (2005)
Acta Arithmetica
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Jing Guo, Xiaoxue Li (2016)
Czechoslovak Mathematical Journal
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For any positive integer , it is easy to prove that the -polygonal numbers are . The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet -functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums for -polygonal numbers with , and give an interesting computational formula for it.
Kaisa Matomäki (2011)
Bulletin de la Société Mathématique de France
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We prove that the sign of Kloosterman sums changes infinitely often as runs through the square-free numbers with at most prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.
Arya Chandran, Neha Elizabeth Thomas, K. Vishnu Namboothiri (2022)
Czechoslovak Mathematical Journal
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Menon’s identity is a classical identity involving gcd sums and the Euler totient function . A natural generalization of is the Klee’s function . We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).
Kathryn E. Hare, Shuntaro Yamagishi (2014)
Acta Arithmetica
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Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.
Han Zhang, Wenpeng Zhang (2015)
Czechoslovak Mathematical Journal
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About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let be a prime, and let denote the number of all such that and . The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function and give an interesting asymptotic formula...
Daniel J. Katz, Philippe Langevin (2015)
Acta Arithmetica
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We consider Weil sums of binomials of the form , where F is a finite field, ψ: F → ℂ is the canonical additive character, , and . If we fix F and d, and examine the values of as a runs through , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...
Zhi-Wei Sun, Mao-Hua Le (2001)
Acta Arithmetica
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Todd Cochrane, Derrick Hart, Christopher Pinner, Craig Spencer (2014)
Acta Arithmetica
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Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain , and for s ≥ 6, . For s ≥ 24 further improvements are made, such as and .
Alexander E. Patkowski (2017)
Czechoslovak Mathematical Journal
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We examine the -Pell sequences and their applications to weighted partition theorems and values of -functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous...
Christoph Aistleitner, István Berkes, Kristian Seip (2015)
Journal of the European Mathematical Society
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Upper bounds for GCD sums of the form are established, where is any sequence of distinct positive integers and ; the estimate for solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for . The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to...
Johannes F. Morgenbesser (2011)
Acta Arithmetica
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Mohammed Berkani, Mustapha Sarih, Hassan Zariouh (2016)
Mathematica Bohemica
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We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties , , and are not preserved under direct sums of operators. However, we prove that if and are bounded linear operators acting on Banach spaces and having the property , then has the property if and only if , where is the upper semi-B-Weyl spectrum of . We obtain analogous preservation results for the properties ,...
Jian-Ping Fang (2016)
Czechoslovak Mathematical Journal
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We derive two identities for multiple basic hyper-geometric series associated with the unitary group. In order to get the two identities, we first present two known -exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two -Chu-Vandermonde summations established by Milne, we arrive at our results. Using the identities obtained in this paper, we give two interesting identities involving binomial...
Sergei V. Astashkin, Lech Maligranda (2010)
Studia Mathematica
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The Rademacher sums are investigated in the Cesàro spaces (1 ≤ p ≤ ∞) and in the weighted Korenblyum-Kreĭn-Levin spaces on [0,1]. They span l₂ space in for any 1 ≤ p < ∞ and in if and only if the weight w is larger than on (0,1). Moreover, the span of the Rademachers is not complemented in for any 1 ≤ p < ∞ or in for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l₂, this span is...
Humio Ichimura (2013)
Acta Arithmetica
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Let be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order (resp. order dividing q), and let ψₙ be an even character of conductor and order pⁿ. We put χ = δφψₙ, whose value is contained in . It is well known that the Bernoulli number is not zero, which is shown in an analytic way. In the extreme cases and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: for any...
Albert Baernstein II, Robert C. Culverhouse (2002)
Studia Mathematica
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Let , , where the are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and are real constants. We prove that if is majorized by in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...
S. Gurak (2007)
Acta Arithmetica
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Swami Jnanananda (1936)
Časopis pro pěstování matematiky a fysiky
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