Gaussian integer points of functions analytic in a half-plane

Fletcher, A. (2008) Gaussian integer points of functions analytic in a half-plane. Mathematical Proceedings of the Cambridge Philosophical Society, 145(2), pp. 257-272. (doi: 10.1017/S0305004108001643)

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: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Fletcher, Dr Alastair
• Authors: Fletcher, A.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Mathematical Proceedings of the Cambridge Philosophical Society
• Journal Abbr.: Proc. Cam. Phil. Soc.
• ISSN: 0305-0041
• ISSN (Online): 1469-8064
• Published Online: 11 June 2008