It has some rather large terms in the continued fraction: [0; 3, 1, 8, 4, 24, 2, 1, 4, 7, 2, 1, 1, 1, 3, 32, 1, 7, 1, 21, 3, 4, 1, 1, 2, 1, 3, 75, 4, 2, 4, 1, 1, 2, 2, 9, 1, ...] Truncating before the 24 gives 37/144. On Tue, Dec 23, 2014 at 1:54 PM, Daniel Asimov <asimov@msri.org> wrote:
Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
* let d_Amax := the real dimension d where the formula
A(d) = 2 pi^(d/2) / Gamma(d/2)
for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum.
-and-
* let d_Vmax := the real dimension d where the formula
V(d) = pi^(d/2) / Gamma(d/2 + 1)
for the d-dimensional content of the unit d-ball in R^d takes its maximum
Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+.
In particular they have the same fractional part:
upsilon := 0.256946404860576780132838388690769236619+
QUESTION: Is anything known about the number theoretic properties of upsilon?
Is it known to be irrational or transcendental? Or related to other numbers, like Euler gamma, whose number-theoretic properties are unknown?
--Dan
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