Gelfond's constant

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is $e π$ , that is, $e$ raised to the power $π$ . Like both $e$ and $π$ , this constant is both an irrational and transcendental number. This was first established by Gelfond and may now be considered as 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 constant was mentioned in Hilbert's seventh problem.^[1] A related constant is $2 \sqrt 2$ , known as the Gelfond–Schneider constant. The related value $π$ + $e π$ is also irrational.^[2]

Numerical value

The decimal expansion of Gelfond's constant begins

e^{\pi }=

23.1406926327792690057290863679485473802661062426002119934450464095243423506904527835169719970675492196... OEIS: A039661

Construction

If one defines $k0 = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/√2⁠$ and

$k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}$

for $n > 0$ , then the sequence^[3]

$(4/k_{n+1})^{2^{-n}}$

converges rapidly to $e π$ .

Continued fraction expansion

$e^{\pi }=23+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{591+{\cfrac {1}{2+{\cfrac {1}{9+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}}}}}$

This is based on the digits for the simple continued fraction:

$e^{\pi }=[23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,1,4,1,2,108,2,2,1,3,1,7,1,2,2,2,1,2,3,2,166,1,2,1,4,8,10,1,1,7,1,2,3,566,1,2,3,3,1,20,1,2,19,1,3,2,1,2,13,2,2,11,...]$

As given by the integer sequence A058287.

Geometric property

The volume of the n-dimensional ball (or n-ball), is given by

$V_{n}={\frac {\pi ^{\frac {n}{2}}R^{n}}{\Gamma \left({\frac {n}{2}}+1\right)}},$

where $R$ is its radius, and $Γ$ is the gamma function. Any even-dimensional ball has volume

$V_{2n}={\frac {\pi ^{n}}{n!}}R^{2n},$

and, summing up all the unit-ball ( $R = 1$ ) volumes of even-dimension gives^[4]

$\sum _{n=0}^{\infty }V_{2n}(R=1)=e^{\pi }.$

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 $e π - π$ , $e π \sqrt 163$ is very close to an integer:

e^{\pi {\sqrt {163}}}=

262537412640768743.9999999999992500725971981856888793538563373369908627075374103782106479101186073129...

\approx 640\,320^{3}+744

This number was discovered in 1859 by the mathematician Charles Hermite.^[5] In a 1975 April Fool article in Scientific American magazine,^[6] "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 $O (e - π \sqrt 163)$ 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 $e π \sqrt 163$ 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 decimal expansion of $e π - π$ is given by A018938:

e^{\pi }-\pi =

19.9990999791894757672664429846690444960689368432251061724701018172165259444042437848889371717254321516...

This is approximately equal to:

19.99910

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.^[7]

The number $π e$

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

\pi ^{e}=

22.4591577183610454734271522045437350275893151339966922492030025540669260403991179123185197527271430315...

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 $a \neq 0$ , $a \neq 1$ ).

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 make no conclusion about the transcendence of $π e$ .

The number $e π - π e$

As with $π e$ , it is not known whether $e π - π e$ is transcendental. Further, no proof exists to show whether or not it is irrational.

The decimal expansion for $e π - π e$ is given by A063504:

e^{\pi }-\pi ^{e}=

0.6815349144182235323019341634048123526767911086035197442420438554574163102913348711984522443404061881...

The number $i i$

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}=

0.2078795763507619085469556198349787700338778416317696080751358830554198772854821397886002778654260353...

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 $x 2 + 1 = 0$ ), and not rational, hence $i i$ is transcendental.

References

^ 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.
^ Nesterenko, Y (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences, Série I. 322 (10): 909–914. Zbl 0859.11047.
^ 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.
^ Connolly, Francis. University of Notre Dame^{[full citation needed]}
^ Barrow, John D (2002). The Constants of Nature. London: Jonathan Cape. p. 72. ISBN 0-224-06135-6.
^ Gardner, Martin (April 1975). "Mathematical Games". Scientific American. 232 (4). Scientific American, Inc: 127. Bibcode:1975SciAm.232e.102G. doi:10.1038/scientificamerican0575-102.
^ Eric Weisstein, "Almost Integer" at MathWorld

External links

[tijdeman-1] 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.

[2] Nesterenko, Y (1996). "Modular Functions and Transcendence Problems". Comptes Rendus de l'Académie des Sciences, Série I. 322 (10): 909–914. Zbl 0859.11047.

[3] 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.

[4] Connolly, Francis. University of Notre Dame^{[full citation needed]}

[5] Barrow, John D (2002). The Constants of Nature. London: Jonathan Cape. p. 72. ISBN 0-224-06135-6.

[6] Gardner, Martin (April 1975). "Mathematical Games". Scientific American. 232 (4). Scientific American, Inc: 127. Bibcode:1975SciAm.232e.102G. doi:10.1038/scientificamerican0575-102.

[MathWorld-7] Eric Weisstein, "Almost Integer" at MathWorld

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Gelfond's constant

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