The Riemann Ξ-function under repeated differentiation

Haseo Ki

doi:10.1016/j.jnt.2005.11.004

The Riemann Ξ-function under repeated differentiation

Haseo Ki

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

Differentiation causes the small gaps between zeros of a given real entire function with order 1 to become larger and the larger gaps to become smaller. In this article, we show that for the Riemann Ξ-function, there exist 〈 A_n 〉 and 〈 C_n 〉 with C_n → 0 such thatunder(lim, n → ∞) A_n Ξ^{(2 n)} (C_n z) = cos z uniformly on compact subsets of C. With our method, one can prove the same result for the analogues of the Riemann Ξ-function from automorphic L-functions. For some other Fourier transforms, we have the similar results as the Riemann Ξ-function.

Original language	English
Pages (from-to)	120-131
Number of pages	12
Journal	Journal of Number Theory
Volume	120
Issue number	1
DOIs	https://doi.org/10.1016/j.jnt.2005.11.004
Publication status	Published - 2006 Sept

Bibliographical note

Funding Information:
1 This work was supported by grant No. R01-2005-000-10339-0 from the Basic Research Program of the Korea Science & Engineering Foundation.

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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10.1016/j.jnt.2005.11.004

Cite this

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abstract = "Differentiation causes the small gaps between zeros of a given real entire function with order 1 to become larger and the larger gaps to become smaller. In this article, we show that for the Riemann Ξ-function, there exist 〈 An 〉 and 〈 Cn 〉 with Cn → 0 such thatunder(lim, n → ∞) An Ξ(2 n) (Cn z) = cos z uniformly on compact subsets of C. With our method, one can prove the same result for the analogues of the Riemann Ξ-function from automorphic L-functions. For some other Fourier transforms, we have the similar results as the Riemann Ξ-function.",

author = "Haseo Ki",

note = "Funding Information: 1 This work was supported by grant No. R01-2005-000-10339-0 from the Basic Research Program of the Korea Science & Engineering Foundation.",

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N2 - Differentiation causes the small gaps between zeros of a given real entire function with order 1 to become larger and the larger gaps to become smaller. In this article, we show that for the Riemann Ξ-function, there exist 〈 An 〉 and 〈 Cn 〉 with Cn → 0 such thatunder(lim, n → ∞) An Ξ(2 n) (Cn z) = cos z uniformly on compact subsets of C. With our method, one can prove the same result for the analogues of the Riemann Ξ-function from automorphic L-functions. For some other Fourier transforms, we have the similar results as the Riemann Ξ-function.

AB - Differentiation causes the small gaps between zeros of a given real entire function with order 1 to become larger and the larger gaps to become smaller. In this article, we show that for the Riemann Ξ-function, there exist 〈 An 〉 and 〈 Cn 〉 with Cn → 0 such thatunder(lim, n → ∞) An Ξ(2 n) (Cn z) = cos z uniformly on compact subsets of C. With our method, one can prove the same result for the analogues of the Riemann Ξ-function from automorphic L-functions. For some other Fourier transforms, we have the similar results as the Riemann Ξ-function.

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The Riemann Ξ-function under repeated differentiation

Abstract

Bibliographical note

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