The sum of digits of
Johannes F. Morgenbesser (2011)
Acta Arithmetica
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Johannes F. Morgenbesser (2011)
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.
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 .
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.
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...
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...
Caroline Sweezy (2004)
Studia Mathematica
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For two strictly elliptic operators L₀ and L₁ on the unit ball in ℝⁿ, whose coefficients have a difference function that satisfies a Carleson-type condition, it is shown that a pointwise comparison concerning Lusin area integrals is valid. This result is used to prove that if L₁u₁ = 0 in B₁(0) and then lies in the exponential square class whenever L₀ is an operator so that L₀u₀ = 0 and implies is in the exponential square class; here S is the Lusin area integral. The exponential...
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.
Charles N. Moore, Xiaojing Zhang (2014)
Studia Mathematica
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We prove a lower bound in a law of the iterated logarithm for sums of the form where f satisfies certain conditions and the satisfy the Hadamard gap condition .
Esteban Andruchow (2003)
Studia Mathematica
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Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), , is continuous but not differentiable. The same holds for the Cayley transform . We also show that the unitary group with the strong operator topology is not an...
Christian Mauduit, Joël Rivat (2015)
Journal of the European Mathematical Society
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For a large class of digital functions , we estimate the sums (and , where denotes the von Mangoldt function (and the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.
Mohamed Amine Hachani (2017)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let be a polynomial of degree having no zeros in , , and let . It was shown by Govil that if and are attained at the same point of the unit circle , then The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.
Z. Knap (1970)
Annales Polonici Mathematici
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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...
Caisheng Chen, Hongxue Song (2016)
Applications of Mathematics
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In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation where is the -Laplacian operator, is continuous and behaves as when . Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution with as is established.