“Masser-Gramain Constant”的概念、定义、习题解题方法及翻译-数学辞典

单词

Masser-Gramain Constant

释义

Masser-Gramain Constant

N.B. A detailed on-line essay by S. Finchwas 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)

where

(2)

is the Supremum, then

is a Polynomial. Furthermore,

is the best constant (i.e., counterexamplesexist for every smaller value).

If is an Entire Function with a Gaussian Integer for each Gaussian Integer , thenGelfond (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)

Maser (1980) proved the weaker result that

must be a Polynomial if

(5)

where

(6)

is the Euler-Mascheroni Constant,

is the Dirichlet Beta Function,

(7)

and

is the minimum Nonnegative

for which there exists a Complex Number

for which the ClosedDisk with center

and radius

contains at least

distinct Gaussian Integers. Gosper gave

(8)

Gramain and Weber (1985, 1987) have obtained

(9)

which implies

(10)

Gramain (1981, 1982) conjectured that

(11)

which would imply

(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.

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