The Riemann Ξ-function under repeated differentiation

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8 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)120-131
Number of pages12
JournalJournal of Number Theory
Issue number1
Publication statusPublished - 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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