Gelfond's constant

In mathematics, Gelfond's constant, named after Aleksander Gelfond, is defined as e^π or $\exp(\pi )$, that is e raised to the power π.

The decimal expansion of Gelfond's constant is given by:

$e^{\pi }=$ 23.14069263277926900572908636794854738026610624260021... (sequence A039661 in the OEIS)

Like both e and π, this constant is both irrational and transcendental. This was first established by Gelfond and may be considered an application of the Gelfond–Schneider theorem, noting that$$e^{\pi }=(e^{i\pi })^{-i}=(-1)^{-i},$$where i is the imaginary unit. Since −i is algebraic but not rational, e^π is transcendental. The numbers π and e^π are also known to be algebraically independent over the rational numbers. Gelfond's constant was mentioned in Hilbert's seventh problem. A related constant is 2^√2, known as the Gelfond–Schneider constant.

Properties

If one defines k₀ = ⁠1/√2⁠ and$$k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}$$for n > 0, then the sequence$$(4/k_{n+1})^{2^{-n}}$$converges rapidly to e^π.

The constant e^π is also related to the volumes of the n-dimensional ball (or n-ball), given by$$V_{n}(R)={\frac {\pi ^{\frac {n}{2}}R^{n}}{\Gamma \left({\frac {n}{2}}+1\right)}},$$where R is its radius, and Γ is the gamma function. Any even-dimensional ball has volume$$V_{2n}(R)={\frac {\pi ^{n}}{n!}}R^{2n},$$and, summing up all even-dimensional unit-ball (R = 1) volumes gives$$\sum _{n=0}^{\infty }V_{2n}(1)=e^{\pi }.$$

Similar or related constants

Ramanujan's constant

$$e^{\pi {\sqrt {163}}}=({\text{Gelfond's constant}})^{\sqrt {163}}$$

This is known as Ramanujan's constant. It is an application of Heegner numbers, where 163 is the Heegner number in question.

Similar to e^π - π, e^π√163 is very close to an integer:

$e^{\pi {\sqrt {163}}}=$ 262537412640768743.99999999999925007259719818568887935385633733699086... $\approx 640\,320^{3}+744$

This number was discovered in 1859 by the mathematician Charles Hermite. In a 1975 April Fool article in Scientific American magazine, "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.

The coincidental closeness, to within 0.000 000 000 000 75 of the number 640320³ + 744 is explained by complex multiplication and the q-expansion of the j-invariant, specifically:$$j((1+{\sqrt {-163}})/2)=(-640\,320)^{3}$$and,$$(-640\,320)^{3}=-e^{\pi {\sqrt {163}}}+744+O\left(e^{-\pi {\sqrt {163}}}\right)$$where O(e^-π√163) is the error term,$${\displaystyle O\left(e^{-\pi {\sqrt {163}}}\right)=-196\,884/e^{\pi {\sqrt {163}}}\approx -196\,884/(640\,320^{3}+744)\approx -0.000\,000\,000\,000\,75}$$which explains why e^π√163 is 0.000 000 000 000 75 below 640320³ + 744.

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

The number e^π − π

The number e^π − π is also very close to an integer, its decimal expansion being given by A018938:

$e^{\pi }-\pi =$ 19.999099979189475767266442984669044496068936843...

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: $$\sum _{k=1}^{\infty }\left(8\pi k^{2}-2\right)e^{-\pi k^{2}}=1.$$ The first term dominates since the sum of the terms for $k\geq 2$ total $\sim 0.0003436.$ The sum can therefore be truncated to $\left(8\pi -2\right)e^{-\pi }\approx 1,$ where solving for $e^{\pi }$ gives $e^{\pi }\approx 8\pi -2.$ Rewriting the approximation for $e^{\pi }$ and using the approximation for $7\pi \approx 22$ gives $$e^{\pi }\approx \pi +7\pi -2\approx \pi +22-2=\pi +20.$$Thus, rearranging terms gives $e^{\pi }-\pi \approx 20.$ Ironically, the crude approximation for $7\pi$ yields an additional order of magnitude of precision.

The number π^e

The decimal expansion of π^e is given by A059850:

$\pi ^{e}=$ 22.459157718361045473427152204543735027589315133...

It is not known whether or not this number is transcendental. Note that, by Gelfond-Schneider theorem, we can only infer definitively that a^b is transcendental if a is algebraic and b is not rational (a and b are both considered complex numbers, also a ≠ 0, a ≠ 1).

In the case of e^π, we are only able to prove this number transcendental due to properties of complex exponential forms, where π is considered the modulus of the complex number e^π, and the above equivalency given to transform it into (-1)^-i, allowing the application of Gelfond-Schneider theorem.

π^e 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 π^e. However the currently unproven Schanuel's conjecture would imply its transcendence.

The number iⁱ

Using the principal value of the complex logarithm, $$i^{i}=(e^{i\pi /2})^{i}=e^{-\pi /2}=(e^{\pi })^{-1/2}$$The decimal expansion of is given by A049006:

$i^{i}=$ 0.207879576350761908546955619834978770033877841...

Because of the equivalence, we can use the Gelfond-Schneider theorem to prove that the reciprocal square root of Gelfond's constant is also transcendental:

i is both algebraic (a solution to the polynomial x² + 1 = 0), and not rational, hence iⁱ is transcendental.

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, ISBN 978-0-521-88268-2

External links

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