Is $\pi/e$ a period?

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

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

By a period I mean

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

See What is ... a period

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.]

Solution 1:

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.