Masser-Gramain Constant

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

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

$$\limsup_{r\to\infty} {\ln M_r\over r}<\ln 2=0.693\ldots,$$ (1)

where

$$M_r=\sup_{\vert z\vert\leq r} \vert f(x)\vert$$ (2)

is the Supremum, then $f$ is a Polynomial. Furthermore, $\ln 2$ is the best constant (i.e., counterexamples exist for every smaller value).

If $f(z)$ is an Entire Function with $f(n)$ a Gaussian Integer for each Gaussian Integer $n$, then Gelfond (1929) proved that there exists a constant $\alpha$ such that

$$\limsup_{r\to\infty} {\ln M_r\over r^2}<\alpha$$ (3)

implies that $f$ is a Polynomial. Gramain (1981, 1982) showed that the best such constant is

$$\alpha={\pi\over 2e}=0.578\ldots.$$ (4)

Maser (1980) proved the weaker result that $f$ must be a Polynomial if

$$\limsup_{r\to\infty} {\ln M_r\over r^2}<\alpha_0={\textstyle... ...\mathop{\rm exp}\nolimits \left({-\delta+{4c\over\pi}}\right),$$ (5)

where

$$c=\gamma\beta(1)+\beta'(1)=0.6462454398948114\ldots,$$ (6)

$\gamma$ is the Euler-Mascheroni Constant, $\beta(z)$ is the Dirichlet Beta Function,

$$\delta\equiv \lim_{n\to\infty} \left({\,\sum_{k=2}^n {1\over\pi{r_k}^2}-\ln n}\right),$$ (7)

and $r_k$ is the minimum Nonnegative $r$ for which there exists a Complex Number $z$ for which the Closed Disk with center $z$ and radius $r$ contains at least $k$ distinct Gaussian Integers. Gosper gave

$$c=\pi\{-\ln[\Gamma({\textstyle{1\over 4}})]+{\textstyle{3\ov... ...pi+{\textstyle{1\over 2}}\ln 2+{\textstyle{1\over 2}}\gamma\}.$$ (8)

Gramain and Weber (1985, 1987) have obtained

$$1.811447299<\delta<1.897327117,$$ (9)

which implies

$$0.1707339<\alpha_0<0.1860446.$$ (10)

Gramain (1981, 1982) conjectured that

$$\alpha_0={1\over 2e},$$ (11)

which would imply

$$\delta=1+{4c\over\pi}=1.822825249\ldots.$$ (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.