The Riemann Ξ-function under repeated differentiation

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5 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 〈 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
Volume120
Issue number1
DOIs
Publication statusPublished - 2006 Sep

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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