Is $\pi/e$ a period?

Question

Is $\displaystyle \frac{\pi}{e}$ a period?

[This was inspired by question Is there a non-trivial definite integral that values to $\frac{e}{\pi}$? ]

a number that can be expressed as an integral of an algebraic function over an algebraic domain

An algebraic domain is a subset of $\mathbb R^n$ given by polynomial inequalities with rational coefficients.

The algebraic function should be a solution of a polynomial equation with integer coefficients. Increasing the dimension by one, we may as well assume the integrand is a rational function with integer coefficients.

We know that $$ \int_{-\infty}^\infty \frac{\cos(x)}{x^2+1} \, \text{dx} = \frac{\pi}{e} $$

And it is a mysterious fact that many integrals involving elementary functions are periods. What about this one?

[The problem cited goes on to ask about a "simple integral" with result $e/\pi$ as well. Presumably $e/\pi$ is not a period, since $\pi$ is a period, and the product $(e/\pi)\;\pi = e$ is conjecturally not a period.]

If $\pi$ is a period then what algebraic function integrates definitely to give $\pi$? — ə̷̶̸͇̘̜́̍͗̂̄︣͟, Oct 17 '18 at 15:01
@TheSimpliFire: To get $\pi$, integrate the algebraic function $1$ on the algebraic domain: unit disk. — GEdgar, Oct 17 '18 at 15:02
@TheSimpliFire If you like, partially integrate Najib's example to give the single-variable integral $\pi = \int_{-1}^1 2 \sqrt{1 - x^2} dx$. — Travis Willse, Oct 17 '18 at 15:04
Is $1 / \pi$ even a period? If so, then $\pi / e$ being a period would imply $1 / e$ being a period, which you might expect not to be true on similar grounds as the fact that $e$ is suspected not to be one. — Sofie Verbeek, Oct 17 '18 at 20:31

score 6 · Accepted Answer · answered Oct 20 '18 at 18:59

It is expected that $\pi/e$ is not a period, but this is not known, and in general it is very difficult to prove that a number is not a period.

In the paper Exponential motives, Fresán and Jossen formulate a version of the Grothendieck period conjecture for certain integrals involving $e$, and Propsition 12.1.4 says that their conjecture implies that $e$ is transcendental over the ring of periods. In particular, since $\pi$ is a period, their conjecture would imply that $\pi/e$ is not a period.

Is $\pi/e$ a period?

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