*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Let be an Entire Function such that is an Integer for each Positive Integer .
Then Pólya (1915) showed that if

(1) |

(2) |

If is an Entire Function with a Gaussian Integer for each Gaussian Integer , then
Gelfond (1929) proved that there exists a constant such that

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

**References**

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/masser/masser.html

Gramain, F. ``Sur le théorème de Fukagawa-Gel'fond.'' *Invent. Math.* **63**, 495-506, 1981.

Gramain, F. ``Sur le théorème de Fukagawa-Gel'fond-Gruman-Masser.'' *Séminaire Delange-Pisot-Poitou
(Théorie des Nombres), 1980-1981*. Boston, MA: Birkhäuser, 1982.

Gramain, F. and Weber, M. ``Computing and Arithmetic Constant Related to the Ring of Gaussian Integers.''
*Math. Comput.* **44**, 241-245, 1985.

Gramain, F. and Weber, M. ``Computing and Arithmetic Constant Related to the Ring of Gaussian Integers.''
*Math. Comput.* **48**, 854, 1987.

Masser, D. W. ``Sur les fonctions entières à valeurs entières.'' *C. R. Acad. Sci. Paris Sér. A-B* **291**, A1-A4, 1980.

© 1996-9

1999-05-26