Fletcher, Alastair. (2008) Gaussian integer points of analytic functions in a half-plane. Mathematical Proceedings of the Cambridge Philosophical Society, Vol.145 (No.2). pp. 257-272. ISSN 0305-0041

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Abstract

A classical result of Pólya states that 2z is the slowest growing transcendental entire function taking integer values on the non-negative integers. Langley generalised this result to show that 2z is the slowest growing transcendental function in the closed right half-plane Ω = {z xs2208 : Re(z) ≥ 0} taking integer values on the non-negative integers. Let E be a subset of the Gaussian integers in the open right half-plane with positive lower density and let f be an analytic function in Ω taking values in the Gaussian integers on E. Then in this paper we prove that if f does not grow too rapidly, then f must be a polynomial. More precisely, there exists L > 0 such that if either the order of growth of f is less than 2 or the order of growth is 2 and the type is less than L, then f is a polynomial.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Mathematical Proceedings of the Cambridge Philosophical Society
Publisher:	Cambridge University Press
ISSN:	0305-0041
Date:	September 2008
Volume:	Vol.145
Number:	No.2
Number of Pages:	16
Page Range:	pp. 257-272
Identification Number:	10.1017/S0305004108001643
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	EPSRC
Grant number:	RA22AP
URI:	http://wrap.warwick.ac.uk/id/eprint/48353

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