Estimates for k-Dimensional Spherical Summations of Arithmetic Functions of the GCD and LCM

Randell Heyman () and László Tóth ()
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Randell Heyman: University of New South Wales, School of Mathematics and Statistics
László Tóth: University of Pécs, Department of Mathematics

A chapter in Number Theory in Memory of Eduard Wirsing, 2023, pp 157-183 from Springer

Abstract: Abstract Let k ≥ 2 $$k\ge 2$$ be a fixed integer. We consider sums of type ∑ n 1 2 + ⋯ + n k 2 ≤ x F ( n 1 , $$\sum _{n_1^2+\cdots + n_k^2\le x} F(n_1,$$ … , n k ) $$\ldots ,n_k)$$ , taken over the k-dimensional spherical region { ( n 1 , … , n k ) ∈ ℤ k : n 1 2 + ⋯ + n k 2 ≤ x } $$\{(n_1,\ldots ,n_k)\in {\mathbb {Z}}^k: n_1^2+\cdots + n_k^2\le x\}$$ , where F : ℤ k → ℂ $$F:{\mathbb {Z}}^k\to {\mathbb {C}}$$ is a given function. In particular, we deduce asymptotic formulas with remainder terms for the spherical summations ∑ n 1 2 + ⋯ + n k 2 ≤ x f ( ( n 1 , … , n k ) ) $$\sum _{n_1^2+\cdots + n_k^2\le x} f((n_1,\ldots ,n_k))$$ and ∑ n 1 2 + ⋯ + n k 2 ≤ x f ( [ n 1 , … , n k ] ) $$\sum _{n_1^2+\cdots + n_k^2\le x} f([n_1,\ldots ,n_k])$$ , involving the GCD and LCM of the integers n 1 , … , n k $$n_1,\ldots ,n_k$$ , where f : ℕ → ℂ $$f:{\mathbb {N}}\to {\mathbb {C}}$$ belongs to certain classes of functions.

Keywords: Arithmetic function; Greatest common divisor; Least common multiple; Number of integer lattice points in a sphere; Spherical summation; Wintner’s mean value theorem; Asymptotic formula (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-31617-3_11

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DOI: 10.1007/978-3-031-31617-3_11

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