## Masser-Gramain Constant

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)

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)

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 Closed Disk 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.