Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Gaussian integer points of analytic functions in a half-plane

ALASTAIR FLETCHERa1

a1 School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD. e-mail: alastair.fletcher@nottingham.ac.uk

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.

(Received November 23 2007)

(Revised January 24 2008)