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Problem
Let $x \in \mathbb{R_{\geq 0}}$, $\alpha \in (0,1]$ and $p \in \mathbb{N}$. Define $$S(x)=\sum_{k=0}^{p-1} \frac{(e^{\frac{2\pi k}{p}i})^{x+1}}{e^{\frac{2\pi k}{p}i}-\alpha} $$
Does there exists $C>0$ such that $S(x) \geq C$ for all $x \in \mathbb{R_{\geq 0}}$?

I encountered similar summation in a journal article, but that was for a special case. I am trying to generalise the summation as above, if possible.

Note that the summation is always real because it involves summing over conjugate pairs. I tried splitting the sum into conjugate pairs and check whether each pair sums to a positive number. However, I am still not sure on how to proceed with that.

Question:
Is it true? May someone kindly provide proof for that? If not, is there any suitable modification on the assumption so that it holds true?

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