Gelfond's constant explained

In mathematics, the exponential of pi,^[1] also called Gelfond's constant,^[2] is the real number raised to the power .

Its decimal expansion is given by:

= ... Like both and, this constant is both irrational and transcendental. This follows from the Gelfond–Schneider theorem, which establishes to be transcendental, given that is algebraic and not equal to zero or one and is algebraic but not rational. We have$$e^\backslash pi\; =\; (e^)^\; =\; (-1)^,$$where is the imaginary unit. Since is algebraic but not rational, is transcendental. The numbers and are also known to be algebraically independent over the rational numbers, as demonstrated by Yuri Nesterenko.^[3] It is not known whether is a Liouville number.^[4] The constant was mentioned in Hilbert's seventh problem alongside the Gelfond-Schneider constant and the name "Gelfond's constant" stems from soviet mathematician Aleksander Gelfond.^[5]

Occurrences

The constant appears in relation to the volumes of hyperspheres:

The volume of an n-sphere with radius is given by:$$V\_n(R)\; =\; \backslash frac,$$where is the gamma function. Considering only unit spheres yields:$$V\_n(1)\; =\; \backslash frac,$$ Any even-dimensional 2n-sphere now gives:$$V\_(1)\; =\; \backslash frac\; =\; \backslash frac$$summing up all even-dimensional unit sphere volumes and utilizing the series expansion of the exponential function gives:^[6] $$\backslash sum\_^\backslash infty\; V\_\; (1)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash exp(\backslash pi)\; =\; e^\backslash pi.$$We also have:

If one defines and$$k\_\; =\; \backslash frac$$for, then the sequence$$(4/k\_)^$$converges rapidly to .^[7]

Similar or related constants

Ramanujan's constant

The number is known as Ramanujan's constant. Its decimal expansion is given by:

= ...

which suprisingly turns out to be very close to the integer : This is an application of Heegner numbers, where 163 is the Heegner number in question. This number was discovered in 1859 by the mathematician Charles Hermite.^[8] In a 1975 April Fool article in Scientific American magazine,^[9] "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name. Ramanujan's constant is also a transcendental number.

The coincidental closeness, to within one trillionth of the number is explained by complex multiplication and the q-expansion of the j-invariant, specifically:$$j((1+\backslash sqrt)/2)=(-640\backslash ,320)^3$$and,$$(-640\backslash ,320)^3=-e^+744+O\backslash left(e^\backslash right)$$where is the error term,$$$$which explains why is 0.000 000 000 000 75 below .

(For more detail on this proof, consult the article on Heegner numbers.)

The number

The number is also very close to an integer, its decimal expansion being given by:

= ...

The explanation for this seemingly remarkable coincidence was given by A. Doman in September 2023, and is a result of a sum related to Jacobi theta functions as follows: $$\backslash sum\_^\backslash left(8\backslash pi\; k^2\; -2\; \backslash right)\; e^\; =\; 1.$$ The first term dominates since the sum of the terms for

total The sum can therefore be truncated to where solving for gives Rewriting the approximation for and using the approximation for gives $$e^\; \backslash approx\; \backslash pi\; +\; 7\backslash pi\; -\; 2\; \backslash approx\; \backslash pi\; +\; 22-2\; =\; \backslash pi+20.$$Thus, rearranging terms gives Ironically, the crude approximation for yields an additional order of magnitude of precision.^[10]

The number

The decimal expansion of is given by:

...

It is not known whether or not this number is transcendental. Note that, by Gelfond-Schneider theorem, we can only infer definitively whether or not is transcendental if and are algebraic (and are both considered complex numbers).

In the case of, we are only able to prove this number transcendental due to properties of complex exponential forms and the above equivalency given to transform it into, allowing the application of Gelfond-Schneider theorem.

has no such equivalence, and hence, as both and are transcendental, we can not use the Gelfond-Schneider theorem to draw conclusions about the transcendence of . However the currently unproven Schanuel's conjecture would imply its transcendence.^[11]

The number

Using the principal value of the complex logarithm$$i^\; =\; (e^)^i\; =\; e^\; =\; (e^)^$$The decimal expansion of is given by:

...

Its transcendence follows directly from the transcendence of .

See also

• Transcendental number
• Transcendental number theory, the study of questions related to transcendental numbers
• Euler's identity
• Gelfond–Schneider constant

Further reading

• Alan Baker and Gisbert Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs 9, Cambridge University Press, 2007,

External links

• Gelfond's constant at MathWorld
• A new complex power tower identity for Gelfond's constant
• Almost Integer at MathWorld

Notes and References

1. Web site: A039661 - OEIS . 2024-10-27 . oeis.org.
2. Web site: Weisstein . Eric W. . Gelfond's Constant . 2024-10-27 . mathworld.wolfram.com . en.
3. Nesterenko, Y . Yuri Valentinovich Nesterenko . 1996 . Modular Functions and Transcendence Problems . . 322 . 10 . 909–914 . 0859.11047.
4. math/0312440 . Michel . Waldschmidt . Open Diophantine Problems . 2004-01-24.
5. Book: Tijdeman, Robert . Robert Tijdeman . Mathematical Developments Arising from Hilbert Problems . . 1976 . 0-8218-1428-1 . Felix E. Browder . Felix Browder . . XXVIII.1 . 241–268 . On the Gel'fond–Baker method and its applications . 0341.10026.
6. Web site: 2019-05-26 . Sums of volumes of unit spheres . 2024-10-27 . www.johndcook.com . en-US.
7. Book: Borwein . J. . Jonathan Borwein . Mathematics by Experiment: Plausible Reasoning in the 21st Century . Bailey . D. . A K Peters . 2004 . 1-56881-211-6 . Wellesley, MA . 137 . 1083.00001 . limited.
8. Book: Barrow , John D . The Constants of Nature . Jonathan Cape . 72 . 2002 . London . 0-224-06135-6 .
9. Gardner . Martin . Mathematical Games . Scientific American . 232 . 4 . 127 . April 1975 . Scientific American, Inc . 10.1038/scientificamerican0575-102 . 1975SciAm.232e.102G .
10. [Eric Weisstein]
11. Web site: Waldschmidt . Michel . 2021 . Schanuel’s Conjecture: algebraic independence of transcendental numbers .