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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:

  1. 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?

  2. Has Goldbach's claim about the irrationality of $S_f$ for nonintegral rational values of $f$ been settled in other cases?

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I'm suspicious there are (simple) direct proofs for ln2 being irrational (the standard argument for proving e^r is irrational seems simple enough). I did find this cute proof however: math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes3.pdf –  Alex R. Feb 7 '10 at 17:58
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It appears that FC answered the second question, but here is another concrete example. If we let f=1/4, then the integral becomes 4-(pi/2)-ln(8). In general, the linear combination mentioned by FC will always have a nonzero algebraic coefficient on pi, except when p/q=1/2 [and that case was dealt with already]. –  Pace Nielsen Feb 10 '10 at 20:41
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@FC: You should write this as an answer, and claim the bounty. –  Theo Johnson-Freyd Feb 10 '10 at 21:07
    
I agree with Theo's comment. –  Franz Lemmermeyer Feb 11 '10 at 6:12
    
FC's comments 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'd accept his comment as an answer if I could. –  Franz Lemmermeyer Feb 11 '10 at 18:47
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up vote 5 down vote accepted

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).

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