# Can $1/\log(2)$ be represented as a period?

In this article by Zagier-Kontsevich, period is defined as values of integral of a rational function over a domain in $$\mathbb{R}^{n}$$ defined by polynomial inequalities with rational coefficients. For example, $$\pi, \log(2), \zeta(3)$$ are periods, and $$e$$ is conjecturally not a period. The set of periods form a ring (period ring), which is not a field. My question is: is $$\frac{1}{\log(2)}$$ also a period? More generally, how about $$\frac{1}{\log\alpha}$$ for $$\alpha \in \overline{\mathbb{Q}}$$? Note that $$\log\alpha$$ is a period for $$\alpha\in \overline{\mathbb{Q}}$$. I tried to find integral representations of $$1/\log(x)$$, but I can't find a suitable one. Also, if it is not, it would be super hard to prove that it is actually not a period.

• See also this post. – Dietrich Burde May 18 at 18:37
• @DietrichBurde Does that "heuristic" implies that $1/\log 2$ might not be a period? – Seewoo Lee May 18 at 18:50