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A sum on Beatty’s theorem

Let \alpha, \beta be positive irrational numbers such that \displaystyle \frac{1}{\alpha} + \frac{1}{\beta}=1. Evaluate the (pseudo) sum:

\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\lfloor n\alpha\rfloor^2}+\frac{1}{\lfloor n\beta\rfloor^2}\right)

Solution

We are using Beatty’s theorem .In brief, it states that for positive irrational numbers \alpha, \beta with \displaystyle \frac{1}{\alpha}+\frac{1}{\beta}=1 the sequences \lfloor \alpha\rfloor, \lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor, \dots and \lfloor \beta\rfloor, \lfloor 2\beta\rfloor, \lfloor 3\beta\rfloor, \dots are complementary. (i.e. disjoint and their union is \mathbb{N}). Thus our sum is nothing else than

\displaystyle \zeta(2)=\frac{\pi^2}{6}

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