Erdős: Sum of rational function of positive integers is either rational or transcendental

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.

Solution 1:

I was not able to find a statement of the conjecture by Erdős and Straus in the form stated, but I did find some interesting related work. I recommend that you take a look at the introduction and bibliography (at least) of the paper

• N. Saradha, R. Tijdeman, "On the transcendence of infinite sums of values of rational functions", J. London Math. Soc. (2) 67 (2003), no. 3, 580-592.

Any paper referenced below without a full citation is in the bibliography of Saradha and Tijdeman. This bibliography may provide a useful starting point in trying to locate the conjecture you mention.

The introduction discusses variations on the following conjecture of Erdős, which apparently goes back to 1949 but did not appear in print in Erdős's intended form until a 1965 paper of Livingston.

Conjecture (Erdős). Let $f$ be a number-theoretic function with period $q > 0$ such that $|f(n)| = 1$ for $1\le n < q$ and $f(q) = 0$. Then $$ \sum_{n=1}^{\infty} \frac{f(n)}{n} \ne 0 $$ whenever the series converges.

In 1973, Baker, Birch, and Wirsing attribute the following problem to Chowla:

Problem. Does there exist a rational-valued function $f(n)$, periodic with prime period $p$, such that $$ \sum_{n=1}^{\infty} \frac{f(n)}{n} = 0? $$

Baker, Birch, and Wirsing disprove the claim by proving the following theorem.

Theorem. Suppose $f\colon \mathbb{Z}\to \overline{\mathbb{Q}}$ is a nonvanishing function with period $q$. If (i) $f(n) = 0$ whenever $1 < \gcd(r, q) < q$, and (ii) the $q$th cyclotomic polynomial is irreducible over $\mathbb{Q}(f(1),\dots,f(q))$, then $$ \sum_{n=1}^{\infty} \frac{f(n)}{n} \ne 0.$$

This result and a related result of Okada are used as a basis for Adhikari, Saradha, Shorey, and Tijdeman to prove the following theorem.

Theorem. Suppose $f\colon \mathbb{Z}\to \overline{\mathbb{Q}}$ is periodic with period $q$. If the series $$ \sum_{n=1}^{\infty} \frac{f(n)}{n} $$ converges to some number $S$, then either $S = 0$ or $S$ is transcendental.

Adhikari et al. comment (I'm paraphrasing here) that applying Baker's theorem tends to lead to dichotomies of the form "$S$ is either rational or transcendental"; beyond that I don't find direct motivation for the claim or prior conjectures of the form you indicate. (But I'm no number theorist.)

Some salt should be ingested with some of the results in Adhikari et al. In particular, I found the following paper which in its introduction claims to provide a counterexample to one of the theorems from that paper:

• M. Genčev, "Transcendence of certain infinite sums involving rational functions", Acta Math. Univ. Ostrav. 15 (2007), no., 1, 7-14.

Solution 2:

There is an obituary of Straus with complete publications at

http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1102706434&page=record

and a list of his Ph.D. students. I count 18 papers with Erdos. Some of them can be downloaded. Not the students, the papers.

Solution 3:

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.