On sum-sets and product-sets of complex numbers

József Solymosi

Displaying similar documents to “On sum-sets and product-sets of complex numbers”

A note on signs of Kloosterman sums

Kaisa Matomäki (2011)

Bulletin de la Société Mathématique de France

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We prove that the sign of Kloosterman sums $Kl (1, 1; n)$ changes infinitely often as $n$ runs through the square-free numbers with at most $15$ 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.

Zero-sums of length kq in $ℤ_{q}^{d}$

Silke Kubertin (2005)

Acta Arithmetica

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On sums of powers of the positive integers

A. Schinzel (2013)

Colloquium Mathematicae

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The pairs (k,m) are studied such that for every positive integer n we have $1^{k} + 2^{k} + \dots + n^{k} | 1^{k m} + 2^{k m} + \dots + n^{k m}$ .

A generalization of a theorem of Erdős-Rényi to m-fold sums and differences

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 $r_{N}^{(m)} (ω)$ 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 $r_{N}^{(m)} (ω) < K$ 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.

On the $k$ -polygonal numbers and the mean value of Dedekind sums

Jing Guo, Xiaoxue Li (2016)

Czechoslovak Mathematical Journal

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For any positive integer $k \geq 3$ , it is easy to prove that the $k$ -polygonal numbers are $a_{n} (k) = (2 n + n (n - 1) (k - 2)) / 2$ . The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet $L$ -functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S (a_{n} (k) {\bar{a}}_{m} (k), p)$ for $k$ -polygonal numbers with $1 \leq m, n \leq p - 1$ , and give an interesting computational formula for it.

Prime numbers along Rudin–Shapiro sequences

Christian Mauduit, Joël Rivat (2015)

Journal of the European Mathematical Society

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For a large class of digital functions $f$ , we estimate the sums $\sum_{n \leq x} Λ (n) f (n)$ (and $\sum_{n \leq x} μ (n) f (n)$ , where $Λ$ denotes the von Mangoldt function (and $μ$ 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.

The sum of digits of $⌊ n^{c} ⌋$

Johannes F. Morgenbesser (2011)

Acta Arithmetica

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Some new sums related to D. H. Lehmer problem

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 $p$ be a prime, and let $N (k; p)$ denote the number of all $1 \leq a_{i} \leq p - 1$ such that $a_{1} a_{2} \dots a_{k} \equiv 1 mod p$ and $2 ∣ a_{i} + {\bar{a}}_{i} + 1,$ $i = 1, 2, \dots, k$ . 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 $N (k; p),$ and give an interesting asymptotic formula...

GCD sums from Poisson integrals and systems of dilated functions

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 $\sum_{k, ℓ = 1}^{N} \frac{{(\gcd (n_{k}, n_{ℓ}))}^{2 α}}{{(n_{k} n_{ℓ})}^{α}}$ are established, where ${(n_{k})}_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0 < α \leq 1$ ; the estimate for $α = 1 / 2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $α = 1 / 2$ . 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...

On the 4-norm of an automorphic form

Valentin Blomer (2013)

Journal of the European Mathematical Society

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We prove the optimal upper bound $\sum_{f} {∥f∥}_{4}^{4} ≪ q^{ϵ}$ where $f$ runs over an orthonormal basis of Maass cusp forms of prime level $q$ and bounded spectral parameter.

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

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We consider Weil sums of binomials of the form $W_{F, d} (a) = \sum_{x \in F} ψ (x^{d} - a x)$ , where F is a finite field, ψ: F → ℂ is the canonical additive character, $g c d (d, | F^{\times} |) = 1$ , and $a \in F^{\times}$ . If we fix F and d, and examine the values of $W_{F, d} (a)$ as a runs through $F^{\times}$ , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $| F^{\times} |$ ). 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...

On the $q$ -Pell sequences and sums of tails

Alexander E. Patkowski (2017)

Czechoslovak Mathematical Journal

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We examine the $q$ -Pell sequences and their applications to weighted partition theorems and values of $L$ -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...

On Sums of Squares in $Q^{\frac{1}{2}} (X)$ etc

W. J. Ellison (1970-1971)

Séminaire de théorie des nombres de Bordeaux

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Exponential sums of the form $\sum χ {(x)}^{a x} ζ_{m}^{b x}$

S. Gurak (2007)

Acta Arithmetica

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Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers

Hajime Kaneko, Takeshi Kurosawa, Yohei Tachiya, Taka-aki Tanaka (2015)

Acta Arithmetica

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Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $\prod_{\binom{k = 1}{U_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) / (U_{d^{k}}))$ (i=1,...,m) or $\prod_{\binom{k = 1}{V_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) (V_{d^{k}})$ (i=1,...,m) to be algebraically dependent, where $a_{i}$ are non-zero integers and $U_{n}$ and $V_{n}$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_{1}, . . ., a_{m}$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically...

Waring's number for large subgroups of ℤₚ

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 $t ₄ = p^{22 / 39 + ϵ}$ , $t ₅ = p^{15 / 29 + ϵ}$ and for s ≥ 6, $t ₛ = p^{(9 s + 45) / (29 s + 33) + ϵ}$ . For s ≥ 24 further improvements are made, such as $t_{32} = p^{5 / 16 + ϵ}$ and $t_{128} = p^{1 / 4}$ .

Estimates of $L_{p}$ norms for sums of positive functions

Ilgiz Kayumov (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We present new inequalities of $L_{p}$ norms for sums of positive functions. These inequalities are useful for investigation of convergence of simple partial fractions in $L_{p} (ℝ)$ .

Polynomial quotients: Interpolation, value sets and Waring's problem

Zhixiong Chen, Arne Winterhof (2015)

Acta Arithmetica

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For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p, w} (u)$ are defined by $q_{p, w} (u) \equiv (u^{w} - u^{w p}) / p m o d p$ with $0 \leq q_{p, w} (u) \leq p - 1$ , u ≥ 0, which are generalizations of Fermat quotients $q_{p, p - 1} (u)$ . First, we estimate the number of elements $1 \leq u < N \leq p$ for which $f (u) \equiv q_{p, w} (u) m o d p$ for a given polynomial f(x) over the finite field $_{p}$ . In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each...