Gelfond's constant

After Soviet mathematician Алекса́ндр О́сипович Ге́льфонд (Alexander Osipovich Gelfond).

Gelfond's constant

1. (mathematics) The value e^π (approximately 23.140692632779), known to be transcendental.
   • 2009, Samuel W. Gilbert, The Riemann Hypothesis and the Roots of the Riemann Zeta Function, BookSurge Publishing, page 93,

     Gelfond's constant satisfies the identity

     ${\displaystyle e^{\pi }=(-1)^{-i}}$

     Therefore, the roots of the Riemann zeta function are defined by geometrical constraints of the discrete partial sums of the Dirichlet series terms by continuous and geometrically equivalent envelopes defined by powers of Gelfond's constant.

   • 2007, Julian Havil, Nonplussed!: Mathematical Proof of Implausible Ideas, Princeton University Press, page 143,

     This means that

     ${\displaystyle \sum _{n\ {\text{even}}}V_{n}(1)=\sum _{m=1}^{\infty }V_{2m}(1)=\sum _{m=1}^{\infty }{\frac {\pi ^{m}}{m!}}=e^{\pi }-1}$

     and we have the promised appearance of Gelfond's constant.

   • 2016, Ravi P. Agarwal, Hans Agarwal, Syamal K. Sen, Birth, growth and computation of pi to ten trillion digits, David H. Bailey, Jonathan M. Borwein (editors, Pi: The Next Generation, Springer, page 403,

     Alexander Osipovich Gelfond (1906-1968) was a Soviet mathematician. He proved that e^π (Gelfond's constant) is transcendental, but nothing yet is known about the nature of the numbers π + e, πe, or π^e.

• Gelfond–Schneider theorem on Wikipedia.Wikipedia
• Gelfond–Schneider constant on Wikipedia.Wikipedia