11 A dichotomy for the dilogarithm at one half
Two statements about classical constants from the library’s work on Diaz’s conjecture. Both rest on the four exponentials theorem in transcendence degree one (Chapter 6). Each alternative of Corollary 11.2 is open, so at least one of two open questions has a positive answer.
Let \(t\neq 0\) be real with \(e^{t}\) algebraic, and let \(a,b,c\in \mathbb {Q}\) with \(a\neq 0\) and \(at^{2}+b\pi ^{2}=c\). Then \(e^{i\gamma /\pi }\) is transcendental for every \(\gamma \in \mathbb {Q}\), \(\gamma \neq 0\).
Source: Brownawell’s Corollary 5 [ Bro74 , p. 23 ] at \(\eta =t/\pi \), followed by rational scaling. The statement for general \(a,b\) was not found in the sources read.
Suppose \(e^{i\gamma /\pi }\) is algebraic, so that \(\lambda =i\gamma /\pi \) is a logarithm of an algebraic number. The relation gives \(t^{2}/(i\pi )=-\frac{c}{a\gamma }\, \lambda +\frac{b}{a}\, i\pi \), again a logarithm of an algebraic number. The matrix \(\begin{pmatrix} t & t^{2}/(i\pi ) \\ i\pi & t \end{pmatrix}\) has determinant \(0\) and entries in a field of transcendence degree one, since \(t\) is algebraic over \(\mathbb {Q}(\pi )\). By Theorem 6.31 its rows or its columns are linearly dependent over \(\mathbb {Q}\); either way \(\alpha t+\beta i\pi =0\) with \(\alpha ,\beta \in \mathbb {Q}\) not both zero, impossible since \(t\) is real and non-zero and \(i\pi \) is purely imaginary.
Either \(\pi ^{2}/12-(\log 2)^{2}/2\) is irrational, or \(e^{i\gamma /\pi }\) is transcendental for every \(\gamma \in \mathbb {Q}\), \(\gamma \neq 0\).
Source: By Euler’s identity, \(\pi ^{2}/12-(\log 2)^{2}/2=\mathrm{Li}_2(1/2)=\sum _{n\ge 1}1/(n^{2}2^{n})\). Its irrationality is listed as unknown in [ Wal04 , p. 274 ] and is still open: [ Hat93 , RV05 , CDT24 ] cover \(\mathrm{Li}_2(1/q)\) only for \(q\ge 6\) and \(q\le -5\). The dichotomy was not found in the sources read.
If \(\pi ^{2}/12-(\log 2)^{2}/2=s\in \mathbb {Q}\), apply Proposition 11.1 with \(t=\log 2\), \(a=-1/2\), \(b=1/12\) and \(c=s\).