Gelfond's constant

In mathematics, the exponential of pi eπ, also called Gelfond's constant, is the real number e raised to the power π (i.e., the value of the exponential function at π).

Its decimal expansion is given by

eπ = 23.14069263277926900572... (sequenceA039661in theOEIS).

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 have e π = ( e i π ) − i = ( − 1 ) − i , {\displaystyle e^{\pi }=(e^{i\pi })^{-i}=(-1)^{-i},} where i is the imaginary unit. Since −i is algebraic but not rational, eπ is transcendental. The numbers π, eπ, and the lemniscate constant are also known to be algebraically independent over the rational numbers, as demonstrated by Yuri Nesterenko. It is not known whether eπ is a Liouville number. The constant was mentioned in Hilbert's seventh problem alongside the Gelfond–Schneider constant 2√2, and the name "Gelfond's constant" stems from Soviet mathematician Alexander Gelfond.

Occurrences

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

The volume of an n-sphere with radius R is given by V n ( R ) = π n 2 R n Γ ( n 2 + 1 ) , {\displaystyle V_{n}(R)={\frac {\pi ^{\frac {n}{2}}R^{n}}{\Gamma \left({\frac {n}{2}}+1\right)}},} where Γ is the gamma function. Considering only unit spheres (R = 1) yields V n ( 1 ) = π n 2 Γ ( n 2 + 1 ) , {\displaystyle V_{n}(1)={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}},}

Any even-dimensional 2n-sphere now gives V 2 n ( 1 ) = π n Γ ( n + 1 ) = π n n ! . {\displaystyle V_{2n}(1)={\frac {\pi ^{n}}{\Gamma (n+1)}}={\frac {\pi ^{n}}{n!}}.} Summing up all even-dimensional unit sphere volumes and utilizing the series expansion of the exponential function gives ∑ n = 0 ∞ V 2 n ( 1 ) = ∑ n = 0 ∞ π n n ! = exp ⁡ ( π ) = e π . {\displaystyle \sum _{n=0}^{\infty }V_{2n}(1)=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}=\exp(\pi )=e^{\pi }.}

Also, if one defines k0 = ⁠1/√2⁠ and k n + 1 = 1 − 1 − k n 2 1 + 1 − k n 2 {\displaystyle 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 {\displaystyle (4/k_{n+1})^{2^{-n}}} converges rapidly to eπ.

Similar or related constants

Ramanujan's constant

The number eπ√163 is known as Ramanujan's constant. Its decimal expansion is given by

eπ√163 = 262537412640768743.99999999999925007259... (sequenceA060295in theOEIS),

which 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. 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. 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: j ( ( 1 + − 163 ) / 2 ) = ( − 640 320 ) 3 {\displaystyle j{\big (}(1+{\sqrt {-163}})/2{\big )}=(-640\,320)^{3}} and ( − 640 320 ) 3 = − e π 163 + 744 + O ( e − π 163 ) , {\displaystyle (-640\,320)^{3}=-e^{\pi {\sqrt {163}}}+744+O\left(e^{-\pi {\sqrt {163}}}\right),} where the error term O ( e − π 163 ) = − 196 884 / e π 163 ≈ − 196 884 / ( 640 320 3 + 744 ) ≈ − 0.000 000 000 000 75 , {\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 eπ√163 is 0.00000000000075 below 6403203 + 744.

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

The number e π − π

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

eπ − π = 19.99909997918947576726... (sequenceA018938in theOEIS).

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

The number π e

The decimal expansion of πe is given by

π e = {\displaystyle \pi ^{e}=} 22.45915771836104547342... (sequenceA059850in theOEIS).

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 cannot use the Gelfond–Schneider theorem to draw conclusions about the transcendence of πe. However the currently unproven Schanuel's conjecture would imply its transcendence.

The number i i

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

i i = {\displaystyle i^{i}=} 0.20787957635076190854... (sequenceA049006in theOEIS).

Its transcendence follows directly from the transcendence of eπ and directly from the Gelfond–Schneider theorem.