= Roth criterion for rational Fredholm values
{c}
{title2=$l(1-\log p/\log q)>2$}
For $0<p<q$, truncating the <Fredholm series> at $n$ gives a rational with denominator at most $Q_n=q^{l^n}$ and positive error at most $2Q_n^{-l(1-\log p/\log q)}$ for large $n$. The displayed strict inequality first rules out a rational value by the rational separation bound, and then rules out an algebraic irrational value by <Roth theorem>. Reduced denominators are at most $Q_n$ and tend to infinity because the truncations are distinct. In particular $l\ge3$ and $p<q^\delta$, $\delta<1/3$, imply transcendence.
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