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Masser-Gramain Constant

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The Masser-Gramain constant is the limit

delta=lim_(n->infty)(sum_(k=2)^n1/(pir_k^2)-lnn),
(1)

where r_k is the minimum radius of a closed disk containing at least k distinct Gaussian integers. The center need not be a Gaussian integer.

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->infty)(lnM_r)/r<ln2=0.693...
(2)

(OEIS A002162), where

M_r=sup_(|z|<=r)|f(z)|
(3)

is the supremum, then f is a polynomial. The constant ln2 is sharp: the entire function f(z)=2^z gives a counterexample at the boundary, so the threshold cannot be increased.

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->infty)(lnM_r)/(r^2)<alpha
(4)

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

alpha=pi/(2e)=0.5778...
(5)

(OEIS A086056).

Masser (1980) proved the weaker result that f must be a polynomial if

limsup_(r->infty)(lnM_r)/(r^2)<alpha_0,
(6)

where

alpha_0=1/2exp(-delta+(4c)/pi),
(7)

and

c=gammabeta(1)+beta^'(1)
(8)
=1/4pi{2gamma+2ln2+3lnpi-4ln[Gamma(1/4)]}
(9)
approx 0.64624543989481...,
(10)

(OEIS A086057), where gamma is the Euler-Mascheroni constant, beta(z) is the Dirichlet beta function, and Gamma(z) is the gamma function.

Gramain and Weber (1985), corrected by Gramain (1987), obtained

1.811447299<delta<1.897327117,
(11)

which implies

0.1707339<alpha_0<0.1860446.
(12)

Gramain (1981, 1982) conjectured that

alpha_0=1/(2e),
(13)

which would imply

delta=1+(4c)/pi=1.822825249...
(14)

(OEIS A086058). This is the decimal expansion of the conjectured expression, rather than of delta. Melquiond et al. (2013) disproved the conjecture by establishing the bounds

1.819776<delta<1.819833.
(15)

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References

Finch, S. R. "Masser-Gramain Constant." §7.2 Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 459-461, 2003.Gel'fond, A. O. "Sur un théorème de M. G. Pólya." Atti Accad. Naz. Lincei Rend. 10, 568-574, 1929.Gosper, R. W. "http://www.mathsoft.com/asolve/constant/constant.html." math-fun@cs.arizona.edu posting, Sept. 27, 1996.Gramain, F. "Corrigendum: Computing an Arithmetic Constant Related to the Ring of Gaussian Integers." Math. Comput. 48, 854, 1987.Gramain, F. "Sur le théorème de Fukasawa-Gel'fond." Invent. Math. 63, 495-506, 1981.Gramain, F. "Sur le théorème de Fukasawa-Gel'fond-Gruman-Masser." Séminaire Delange-Pisot-Poitou (Théorie des Nombres), 1980-1981. Boston, MA: Birkhäuser, 1982.Gramain, F. and Weber, M. "Computing an Arithmetic Constant Related to the Ring of Gaussian Integers." Math. Comput. 44, 241-250, 1985.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 291, A1-A4, 1980.Melquiond, G.; Nowak, W. G.; and Zimmermann, P. "Numerical Approximation of the Masser-Gramain Constant to Four Decimal Digits: delta=1.819...." Math. Comput. 82, 1235-1246, 2013. https://doi.org/10.1090/S0025-5718-2012-02635-4.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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Masser-Gramain Constant

Cite this as:

Weisstein, Eric W. "Masser-Gramain Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Masser-GramainConstant.html

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