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Gaussian integer points of analytic functions in a half-plane


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 : 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.

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[1] S. B. Bank and J. K. Langley . On the value distribution theory of elliptic functions. Monatsh. Math. 98 no. 1 (1984), 120.

[4] R. C. Buck . Integral valued entire functions. Duke Math. J. 15 (1948), 879891.

[8] J. K. Langley . Integer points of meromorphic functions. Comput. Methods Funct. Theory 5 (2005), 253262.

[10] J. K. Langley . Integer-valued analytic functions in a half-plane. Comput. Methods Funct. Theory 7 (2007), 433442.

[12] R. P. Robinson . Integer-valued entire functions. Trans. Amer. Math. Soc. 153 (1971), 451468.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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