Masser-Gramain Constant
المؤلف:
Finch, S. R
المصدر:
Masser-Gramain Constant." §7.2 Mathematical Constants. Cambridge, England: Cambridge University Press
الجزء والصفحة:
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23-1-2019
2671
Masser-Gramain Constant
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
(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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