TITLE: Erdős: Sum of rational function of positive integers is either rational or transcendental
QUESTION [14 upvotes]: I am trying to find a conjecture apparently made by Erdős and Straus. I say apparently because I have had so much trouble finding anything information about it that I'm beginning to doubt its existence. Here it is:

Let $\phi(X)$ be a rational function over $\mathbb{Q}$ that is defined at every positive integer. If the sum $\sum \limits_{n=1}^\infty \phi(n)$ converges, it is either rational or transcendental, i.e., it is never an irrational algebraic number.

Has anyone heard of this conjecture? I was told about it by my supervisor, but he doesn't remember where he heard about it.

REPLY [3 votes]: Assume that Erdos theorem is true. Then $f_s(n)=\frac{1}{n^{2s+1}}$, $s=1,2,\ldots$ are clearly a rational functions and
$$
\sum^{\infty}_{n=1}f_s(n)=\sum^{\infty}_{n=1}\frac{1}{n^{2s+1}}=\zeta(2s+1),
$$
will be rationals or transcendentals, which I think is highly considerable statement. Hence I expect the proof to your answer must be quite hard.
Note also that rational function is $\frac{n^2-2}{n^5+1}$ but $F_n$ is not, where $F_n$is the $n-$th Fibonacci number.