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3 questions
13
votes
1
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782
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Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,...$ (where $H(k)$ is the Hamming-weight)
In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...
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5
votes
1
answer
742
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mertens-function in the light of divergent summation - what summation method were best adapted
Just reading about the Mertens-function in the other thread
Mertens function I remember an earlier attempt to apply divergent summation
to the series which is constructed of the Moebius-function
at ...
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-2
votes
1
answer
217
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Convergence and roots of alternating periodic infinite series
Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...
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