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Wikipedia gives an interesting infinite sum for Euler's constant $\gamma$ and I was wondering how one would evaluate this interesting sum. The sum is given as follows:

Let $N_0 (x)$ and $N_1 (x)$ represent the number of zeros (OEIS A023416) and ones (OEIS A000120) respectively of the binary expansion of $n$.

$ \sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} = \gamma $

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    On the linked Wikipedia page it also states Nielsen (1897) found $\gamma= 1- \sum_{k=2}^{\infty} (-1)^k \frac{ \lfloor \log_2 k \rfloor }{k+1} .$ Another way to write $\lfloor \log_2 k \rfloor +1$ is $\lfloor \log_2 2k \rfloor $ so our series could also be viewed as $\gamma = \sum_{k=1}^{\infty} \frac{ \lfloor \log_2 2k \rfloor }{ 2k (2k+1) }.$ This series appears to be related to the even terms of the sum of Vacca's and Nielsen's series.2012-08-16

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