Zero-sums of length kq in
Silke Kubertin (2005)
Acta Arithmetica
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Silke Kubertin (2005)
Acta Arithmetica
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Davide Schipani, Michele Elia (2012)
Bulletin of the Polish Academy of Sciences. Mathematics
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An elementary approach is shown which derives the values of the Gauss sums over , p odd, of a cubic character. New links between Gauss sums over different field extensions are shown in terms of factorizations of the Gauss sums themselves, which are then revisited in terms of prime ideal decompositions. Interestingly, one of these results gives a representation of primes p of the form 6k+1 by a binary quadratic form in integers of a subfield of the cyclotomic field of the pth roots of...
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.
A. Schinzel (2013)
Colloquium Mathematicae
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The pairs (k,m) are studied such that for every positive integer n we have .
József Solymosi (2005)
Journal de Théorie des Nombres de Bordeaux
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We give a simple argument that for any finite set of complex numbers , the size of the the sum-set, , or the product-set, , is always large.
Johannes F. Morgenbesser (2011)
Acta Arithmetica
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S. Gurak (2007)
Acta Arithmetica
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Stéphane Louboutin (2001)
Colloquium Mathematicae
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Let k ≥ 1 denote any positive rational integer. We give formulae for the sums (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.
Matthias Beck, Florian Kohl (2014)
Acta Arithmetica
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We introduce and study the Rademacher-Carlitz polynomial where , s,t ∈ ℝ, and u and v are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view R(u,v,s,t,a,b) as a polynomial analogue (in the sense of Carlitz) of the Dedekind-Rademacher sum , which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz...
W. J. Ellison (1970-1971)
Séminaire de théorie des nombres de Bordeaux
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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.
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.
Valentin Blomer (2013)
Journal of the European Mathematical Society
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We prove the optimal upper bound where runs over an orthonormal basis of Maass cusp forms of prime level and bounded spectral parameter.
Mats Erik Andersson (2001)
Colloquium Mathematicae
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By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.
Andrzej Kłopotowski
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CONTENTSIntroduction.......................................................................................................................................................................... 5 I. Infinitely divisible probability measures on ....................................................................................... 6 II. The classical limit theorems for sums of independent random vectors................................................ 14 III. Convergence in law to ℒ (,...
Davide Schipani, Michele Elia (2011)
Bulletin of the Polish Academy of Sciences. Mathematics
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By an elementary approach, we derive the value of the Gauss sum of a cubic character over a finite field without using Davenport-Hasse’s theorem (namely, if s is odd the Gauss sum is -1, and if s is even its value is ).
Felix L. Schwenninger, Hans Zwart (2015)
Studia Mathematica
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In this paper we show that from an estimate of the form , we can conclude that C(t) equals cos(at)I. Here is a strongly continuous cosine family on a Banach space.