On Sums of Three Squares

Choi, Stephen; Kumchev, Angel V.; Osburn, Robert

doi:10.1142/s1793042105000054

Cited by 9 publications

(12 citation statements)

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“…We have L(s; θ k θ k ′ ) = ∞ n=1 r k (n)r k ′ (n)n −s . Then by applying Corollary 9.3 (ii) to L(s; θ k θ k ), we obtain 0<n≤X r k (n) 2 ∼ π k ζ(k−1) Γ(k/2) 2 ζ(k)(1−2 −k ) X k−1 k−1 for k ≥ 3, which is known as Wagon's conjecture proved by R. Crandall and S. Wagon (for the detail see Borwein and Choi [1], Choi, Kumchev and Osburn [3]). More generally we obtain from Corollary 9.3 (iii), for k ∈ N, m ∈ Z, ≥ 0, k > max{4m, 2}.…”

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confidence: 71%

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Petersson scalar products and L-functions arising from modular forms

Tsuyumine

2019

Ramanujan J

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The Rankin-Selberg method developed by Rankin [8] and by Selberg [10] gives us L-functions from cuspidal automorphic forms by taking the scalar products of them with real analytic Eisenstein series of weight 0. Some of analytic properties of Eisenstein series inherit to the L-functions such as functional equations or positions of possible poles. In Zagier [11], he shows that this important method can be applied to the automorphic forms on SL 2 (Z) which are not cuspidal. The researches in this direction are made also by Gupta [4], Chiera [2].

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confidence: 71%

“…with a Dirichlet character χ modulo N where c, d runs over the set of second rows of matrices in Γ 0 (N ), or Eisenstein series of half integral weight (see Sect. 3). We show that the analytic properties of these Eisenstein series inherit to the following L-functions.…”

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confidence: 85%

Petersson scalar products and L-functions arising from modular forms

Tsuyumine

2019

Ramanujan J

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“…The measures σ Λ and σ S are odd. To check the identity σ Λ ,φ = −ihσ S , ϕi [14] for every test function ϕ it suffices to do it for every odd ϕ. Let ω =φ be the 1D Fourier transform of ϕ.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Let ω =φ be the 1D Fourier transform of ϕ. Then ω is also an odd function in the Schwartz class SðRÞ and the left-hand side of [14] is…”

Section: Proof Of Theoremmentioning

confidence: 99%

Measures with locally finite support and spectrum

Meyer¹

2017

Rev. Mat. Iberoam.

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The goal of this paper is the construction of measures μ on R n enjoying three conflicting but fortunately compatible properties: (i) μ is a sum of weighted Dirac masses on a locally finite set, (ii) the Fourier transformμ of μ is also a sum of weighted Dirac masses on a locally finite set, and (iii) μ is not a generalized Dirac comb. We give surprisingly simple examples of such measures. These unexpected patterns strongly differ from quasicrystals, they provide us with unusual Poisson's formulas, and they might give us an unconventional insight into aperiodic order.Poisson formula | spectrum | almost periodic T he Dirac mass at a ∈ R n is denoted by δ a or δ a ðxÞ. A purely atomic measure is a linear combination μ = P λ∈Λ cðλÞδ λ of Dirac masses where the coefficients cðλÞ are real or complex numbers and P jλj≤R jcðλÞj is finite for every R > 0. Then Λ is a countable set of points of R n . If Λ is closed and if cðλÞ ≠ 0, ∀λ ∈ Λ, then Λ is the support of μ. A subset Λ ⊂ R n is locally finite if Λ ∩ B is finite for every bounded set B. Equivalently Λ can be ordered as a sequence fλ j , j = 1,2, . . .g and λ j tends to infinity with j. A measure μ is a tempered distribution if it has a polynomial growth at infinity in the sense given by Laurent Schwartz in ref. 1. For instance, the measureThe distributional Fourier transformμ of μ is defined by the following condition: hμ, ϕi = hμ,φi shall hold for every test function ϕ belonging to the Schwartz class SðR n Þ. The spectrum S of μ is the (closed) support ofμ. Definition 1: A purely atomic measure μ on R n is a crystalline measure if i) the support Λ of μ is a locally finite set, ii) μ is a tempered distribution, and iii) the distributional Fourier transformμ of μ is also a purely atomic measure that is supported by a locally finite set S.If μ is a crystalline measure, its Fourier transform is also a crystalline measure.Definition 2: A measure μ on R is odd if for every compactly supported continuous function f we haveFor every set E of real numbers, let E + = E ∩ fx > 0g and E − = E ∩ fx < 0g. Let us denote by Q the field of rational numbers. Theorem 1 is proved in this article: Theorem 1. There exists an odd crystalline measure μ on R such that its support Λ and its spectrum S have the following properties: (i) Each finite subset of Λ + is linearly independent over Q and (ii) each finite subset of S + is linearly independent over Q.The spectrum S of μ is an increasing sequence s k , k ∈ Z, of real numbers such that (i) s −k = −s k , ∀k ∈ Z, and (ii) s 1 , s 2 , . . . , s N are linearly independent over Q for each integer N. Theorem 1 implies that P ∞ −∞ bðkÞexpð2πis k xÞ is a sum of Dirac masses on a locally finite set Λ. It is counterintuitive that these incoherent waves bðkÞexpð2πis k xÞ can be piled up in harmony so that their sum yields Dirac masses.Theorem 1 is valid in any dimension as the following proposition shows.Theorem 2. There exists a crystalline measure μ on R n such that i) μ is odd in the last variable x n , ii) the support Λ of μ is the union of ...

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“…Let E * (R) be the error term in the approximation of the number of visible lattice points in the sphere of radius R, E * (R) = # n ∈ Z 3 : n 2 R, gcd(n 1 , n 2 , n 3 …”

Section: Introduction and Notationmentioning

confidence: 99%

Visible lattice points in the sphere

Chamizo

Cristóbal

Ubis

2007

Journal of Number Theory

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The number of visible (primitive) lattice points in the sphere of radius R is well approximated by 4π 3ζ(3) R 3 . We consider an integral expression involving the error term E * (R), which leads to E * (R) = Ω(R(log R) 1/2 ). This is comparable to what is known in the sphere problem. We can avoid the use of the second power moment (which is in this case unknown) by employing an auxiliary trigonometric series correlated to E * (R). This approach to prove Ω-results seems to be new and could be useful in other problems.

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On Sums of Three Squares

Abstract: Let r3(n) be the number of representations of a positive integer n as a sum of three squares of integers. We give two alternative proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of r3(n).

Cited by 9 publications

References 15 publications

Petersson scalar products and L-functions arising from modular forms

Petersson scalar products and L-functions arising from modular forms

Measures with locally finite support and spectrum

Visible lattice points in the sphere

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