Variations of Gelfond's constant $e^\pi$

Question

Gelfond's constant is the transcendental number $e^\pi$ which is approximately equal to:

$$ 23.140692632... $$

Are there any known similar constants (i.e. referred to in literature or used in specific contexts) of the form $e^c$ where:

$c=\pi^2, \pi^3, \ldots$?
$c=\pi k$, where $k$ is a constant (such as $\sqrt{163}$ for Ramanujan's constant)?
$c=\pi f(x)$ for some $x$?

What do you mean with "known similar"? Are you asking for what values of $\alpha$ it is known that $e^{\alpha\pi}$ is trascendental? — ajotatxe, Aug 08 '17 at 08:23
Do you mean constants that are well known and referred to as such in mathematical literature? Otherwise I could construct, say, $e^{\pi(\pi-e)}$ and call it "Linnell's constant". Nobody would use it, but it would be a constant, and it would be named (albeit only by me). — Tez LaCoyle, Aug 08 '17 at 08:25
@Linnell Yes, "known" as in "used in mathematical literature" :) — Klangen, Aug 08 '17 at 08:27
There are a couple of other values of $n$ apart from 163 where $e^{\pi \sqrt{n}}$ is interesting: 43 and 67. See Heegner number. — PM 2Ring, Aug 08 '17 at 09:01
@PM2Ring. Thanks for the link ! It is just fascinating. Cheers. — Claude Leibovici, Aug 08 '17 at 09:05
@T.Linnell. I already used your constant because $\lfloor e^{9\pi(\pi-e)}\rfloor=157760$ which I found to be very useful. — Claude Leibovici, Aug 08 '17 at 09:08
No worries. Curiously, $e^{5\pi}$ is also close to an integer: 6635623.999341... — PM 2Ring, Aug 08 '17 at 09:09

score 3 · Accepted Answer · answered Aug 08 '17 at 10:06

Reading through Brownawell and Waldschmidt, they propose results in these directions which do not rely on the so-called Schanuel's Conjecture. The references are as follows

These papers independently prove results along the following lines.

Let $\alpha$, $\beta$, and $\gamma$ be nonzero complex numbers with $\alpha$ and $\beta$ both irrational. If $e^\gamma$ and $e^{\alpha\gamma}$ are both algebraic numbers, then at least two of the numbers $$\alpha, \beta, \gamma, e^{\beta\gamma}, e^{\alpha\beta\gamma}$$ are algebraically independent over $\mathbb{Q}$.

This raises several interesting consequences:

Taking $\alpha=\beta=e^{-1}, \gamma=e^2$, we see that at least one of $e^e$ and $e^{e^2}$ must be transcendental. This was conjectured by Schneider.
Taking $\alpha=\beta=\gamma$, we see that given any nonzero complex number $\alpha$, at least one of the numbers $e^{\alpha}, e^{\alpha^2}, e^{\alpha^3}$ must be transcendental.
Taking $\alpha = \beta = i/\pi, \gamma=\pi^2$, we see that at least one of the following holds: (i) $e^{\pi^2}$ is transcendental, or (ii) $e$ and $\pi$ are algebraically independent.

So as a partial answer to this question, at least one of $e\pi$ and $e^{\pi^2}$ is transcendental.

Whilst this does not answer fully your question, it may be of decent enough reference to continue your quest for showing the transcendental nature of some of the reuslts you seek.

Thank you for your answer, it is highly interesting. I will read those references as soon as I have time. — Klangen, Aug 08 '17 at 10:16
@Pickle [1] http://www.sciencedirect.com/science/article/pii/0022314X73900449 [2] http://www.sciencedirect.com/science/article/pii/0022314X74900055 — Winther, Aug 08 '17 at 10:22
@Pickle Ah, yes I suppose I didn't mean them to link to an online resource, more that these are the references to the journals within which they live. — , Aug 08 '17 at 10:22

Variations of Gelfond's constant $e^\pi$

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