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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.

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

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