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Date: 4-3-2017
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Let be an entire function such that is an integer for each positive integer . Then Pólya (1915) showed that if
(1) |
(OEIS A002162), where
(2) |
is the supremum, then is a polynomial. Furthermore, is the best constant (i.e., counterexamples exist for every smaller value).
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) |
implies that is a polynomial. Gramain (1981, 1982) showed that the best such constant is
(4) |
(OEIS A086056).
Maser (1980) proved the weaker result that must be a polynomial if
(5) |
where
(6) |
and
(7) |
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(8) |
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(9) |
(OEIS A086057), is the Euler-Mascheroni constant, is the Dirichlet beta function, is the gamma function,
(10) |
is known as the Masser-Gramain constant, and is the minimum nonnegative for which there exists a complex number for which the closed disk with center and radius contains at least distinct Gaussian integers.
Gramain and Weber (1985, 1987) have obtained
(11) |
which implies
(12) |
Gramain (1981, 1982) conjectured that
(13) |
which would imply
(14) |
(OEIS A086058).
REFERENCES:
Finch, S. R. "Masser-Gramain Constant." §7.2 Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 459-461, 2003.
Gosper, R. W. "http://www.mathsoft.com/asolve/constant/constant.html." math-fun@cs.arizona.edu posting, Sept. 27, 1996.
Gel'fond, A. O. "Sur un théorème de M. G. Pólya." Atti Accad. Naz. Lincei Rend. 10, 568-574, 1929.
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.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 116-117, 2003.
Masser, D. W. "Sur les fonctions entières à valeurs entières." C. R. Acad. Sci. Paris Sér. A-B 291, A1-A4, 1980.
Pólya, G. "Über ganzwertige ganze Funktionen." Rend. Circ. Mat. Palermo 40, 1-16, 1915.
Sloane, N. J. A. Sequences A002162/M4074, A086056, A086057, and A086058 in "The On-Line Encyclopedia of Integer Sequences."
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