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Gelfond's constant

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

Its decimal expansion is given by:

eπ = 23.14069263277926900572... (sequence A039661 in the OEIS)

Like both e and π, this constant is both irrational and transcendental. This follows from the Gelfond–Schneider theorem, which establishes ab to be transcendental, given that a is algebraic and not equal to zero or one and b is algebraic but not rational. We havewhere i is the imaginary unit. Since i is algebraic but not rational, eπ is transcendental. The numbers π and eπ are also known to be algebraically independent over the rational numbers, as demonstrated by Yuri Nesterenko.[3] It is not known whether eπ is a Liouville number.[4] The constant was mentioned in Hilbert's seventh problem alongside the Gelfond-Schneider constant 22 and the name "Gelfond's constant" stems from soviet mathematician Aleksander Gelfond.[5]

Occurences

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The constant eπ is also appears in relation to the volumes of hyperspheres.

Graphs of volumes () and surface areas () of n-balls of radius 1.

The volumen of an n-sphere with radius R is given by:where Γ is the gamma function. Considering only unit spheres (R = 1) yields: Any even-dimensional 2n-sphere now gives:summing up all even-dimensional unit sphere volumes and utilizing the series expansion of the exponential function gives:[6]We also have:

If one defines k0 = 1/2 andfor n > 0, then the sequenceconverges rapidly to eπ.[7]

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Ramanujan's constant

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The number eπ163 is known as Ramanujan's constant. Its decimal expansion is given by:

eπ163 = 262537412640768743.99999999999925007259... (sequence A060295 in the OEIS)

which suprisingly turns out to be very close to the integer 6403203 + 744: 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 6403203 + 744 is explained by complex multiplication and the q-expansion of the j-invariant, specifically:and,where O(e-π163) is the error term,which explains why eπ163 is 0.000 000 000 000 75 below 6403203 + 744.

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

The number eππ

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

eππ = 19.99909997918947576726... (sequence A018938 in the OEIS)

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: 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 Thus, rearranging terms gives Ironically, the crude approximation for yields an additional order of magnitude of precision.[10]

The number πe

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The decimal expansion of πe is given by:

22.45915771836104547342... (sequence A059850 in the OEIS)

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 ab is transcendental if a and b are algebraic (a and b are both considered complex numbers).

In the case of eπ, 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 (-1)-i, allowing the application of Gelfond-Schneider theorem.

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

The number ii

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Using the principal value of the complex logarithmThe decimal expansion of is given by:

0.20787957635076190854... (sequence A049006 in the OEIS)

Its transcendence follows directly from the transcendence of eπ.

See also

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References

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  1. ^ "A039661 - OEIS". oeis.org. Retrieved 2024-10-27.
  2. ^ Weisstein, Eric W. "Gelfond's Constant". mathworld.wolfram.com. Retrieved 2024-10-27.
  3. ^ Nesterenko, Y (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences, Série I. 322 (10): 909–914. Zbl 0859.11047.
  4. ^ Waldschmidt, Michel (2004-01-24). "Open Diophantine Problems". arXiv:math/0312440.
  5. ^ Tijdeman, Robert (1976). "On the Gel'fond–Baker method and its applications". In Felix E. Browder (ed.). Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. Vol. XXVIII.1. American Mathematical Society. pp. 241–268. ISBN 0-8218-1428-1. Zbl 0341.10026.
  6. ^ "Sums of volumes of unit spheres". www.johndcook.com. 2019-05-26. Retrieved 2024-10-27.
  7. ^ Borwein, J.; Bailey, D. (2004). Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters. p. 137. ISBN 1-56881-211-6. Zbl 1083.00001.
  8. ^ Barrow, John D (2002). The Constants of Nature. London: Jonathan Cape. p. 72. ISBN 0-224-06135-6.
  9. ^ Gardner, Martin (April 1975). "Mathematical Games". Scientific American. 232 (4). Scientific American, Inc: 127. Bibcode:1975SciAm.232e.102G. doi:10.1038/scientificamerican0575-102.
  10. ^ Eric Weisstein, "Almost Integer" at MathWorld
  11. ^ Waldschmidt, Michel (2021). "Schanuel's Conjecture: algebraic independence of transcendental numbers" (PDF).

Further reading

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