Consider the series
$$ S_f = \sum_{x=1}^\infty \frac{f}{x^2+fx}. $$
Goldbach showed that, for integers $f \ge 1$,
$$ S_f = 1 + \frac12 + \frac13 + \ldots + \frac1f $$
(this follows easily by writing $S_f$ as a telescoping series).
Thus $S_f$ is rational for all natural numbers $f \ge 1$.
Goldbach claimed that, for all nonintegral (rational) numbers $f$,
the sum $S_f$ would be irrational.
Euler showed, by using the substitution
$$ \frac1k = \int_0^1 x^{k-1} dx, $$
that
$$ S_f = \int_0^1 \frac{1-x^f}{1-x} dx. $$
He evaluated this integral for $f = \frac12$ and found
that $S_{1/2} = 2(1 – \ln 2)$ (this also follows easily from
Goldbach's series for $S_f$). Thus Goldbach's claim holds for all
$f \equiv \frac12 \bmod 1$ since $S_{f+1} = S_f + \frac1{f+1}$.
Here are my questions:
-
The irrationality of $\ln 2$ was established by Lambert, who
proved that $e^r$ is irrational for all rational numbers
$r \ne 0$. Are there any (simple) direct proofs? -
Has Goldbach's claim about the irrationality of $S_f$ for
nonintegral rational values of $f$ been settled in other cases?
Best Answer
Please allow me to put my question on top of the list again by turning my comment into an answer. FC's remarks led me to the article "Transcendental values of the digamma function", J. Number Theory 125, No. 2, 298-318 (2007) by Ram Murty and N. Saradha, where Thm. 9 states that the values of S_f are transcendental for rational numbers 0 < f < 1. I apologize for not having asked this question in 2006, which is why I have only a bounty to offer (and a reference to FC from MO in Euler's OO).