On sums of powers of the positive integers

A. Schinzel

Displaying similar documents to “On sums of powers of the positive integers”

Zero-sums of length kq in $ℤ_{q}^{d}$

Silke Kubertin (2005)

Acta Arithmetica

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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.

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.

A Menon-type identity using Klee's function

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 $φ$ . A natural generalization of $φ$ is the Klee’s function $Φ_{s}$ . 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).

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.

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...

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...

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Zhi-Wei Sun, Mao-Hua Le (2001)

Acta Arithmetica

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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 $t_{128} = p^{1 / 4}$ .

A new proof of the $q$ -Dixon identity

Victor J. W. Guo (2018)

Czechoslovak Mathematical Journal

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We give a new and elementary proof of Jackson’s terminating $q$ -analogue of Dixon’s identity by using recurrences and induction.

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...

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...

The sum of digits of $⌊ n^{c} ⌋$

Johannes F. Morgenbesser (2011)

Acta Arithmetica

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