Exponential sums of the form $∑ χ(x)^{ax} ζ_{m}^{bx}$

S. Gurak

Displaying similar documents to “Exponential sums of the form $\sum χ {(x)}^{a x} ζ_{m}^{b x}$ ”

The sum of digits of $⌊ n^{c} ⌋$

Johannes F. Morgenbesser (2011)

Acta Arithmetica

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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.

On a sum involving the integral part function

Bo Chen (2024)

Czechoslovak Mathematical Journal

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Let $[t]$ be the integral part of a real number $t$ , and let $f$ be the arithmetic function satisfying some simple condition. We establish a new asymptotical formula for the sum $S_{f} (x) = \sum_{n \leq x} f ([x / n])$ , which improves the recent result of J. Stucky (2022).

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 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.

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...

Characterization of functions whose forward differences are exponential polynomials

J. M. Almira (2017)

Commentationes Mathematicae Universitatis Carolinae

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Given ${h_{1}, \dots, h_{t}}$ a finite subset of $ℝ^{d}$ , we study the continuous complex valued functions and the Schwartz complex valued distributions $f$ defined on $ℝ^{d}$ with the property that the forward differences $Δ_{h_{k}}^{m_{k}} f$ are (in distributional sense) continuous exponential polynomials for some natural numbers $m_{1}, \dots, m_{t}$ .

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...

Exponential domination in function spaces

Vladimir Vladimirovich Tkachuk (2020)

Commentationes Mathematicae Universitatis Carolinae

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Given a Tychonoff space $X$ and an infinite cardinal $κ$ , we prove that exponential $κ$ -domination in $X$ is equivalent to exponential $κ$ -cofinality of $C_{p} (X)$ . On the other hand, exponential $κ$ -cofinality of $X$ is equivalent to exponential $κ$ -domination in $C_{p} (X)$ . We show that every exponentially $κ$ -cofinal space $X$ has a $κ^{+}$ -small diagonal; besides, if $X$ is $κ$ -stable, then $n w (X) \leq κ$ . In particular, any compact exponentially $κ$ -cofinal space has weight not exceeding $κ$ . We also establish that any exponentially $κ$ -cofinal...

Elliptic functions, area integrals and the exponential square class on B₁(0) ⊆ ℝⁿ, n > 2

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 $S u ₁ \in L^{\infty} (S^{n - 1})$ then ${u ₁ |}_{S^{n - 1}} = f$ lies in the exponential square class whenever L₀ is an operator so that L₀u₀ = 0 and $S u ₀ \in L^{\infty}$ implies ${u ₀ |}_{S^{n - 1}}$ is in the exponential square class; here S is the Lusin area integral. The exponential...

Displaying similar documents to “Exponential sums of the form ∑ χ ( x ) a x ζ m b x ”

Displaying similar documents to “Exponential sums of the form $\sum χ {(x)}^{a x} ζ_{m}^{b x}$ ”