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Maximum of $\sum \frac{y_n}{n^x}$

Let $x$ be real, and $y_n$ be real numbers. Does
$$
M_x= \sup \sum \frac{y_n}{n^x}
$$
exist? And if so, does it have a simple characterization (for example I have seen that it exists whenever $x$ is rational)?

A:

This is a very interesting question!
First, there are a lot of results (implicit or explicit) for the case when $y_n=y$ and $x$ is integer. For example, if $y\in\mathbb{R}$, $y_n=\frac{n^y}{n-1}$, $n\geq 1$, and $x=1$, then
\begin{align*}
M_1&=\sum_{n\geq 1}\frac{y}{n-1}=\sum_{n\geq 1}\left(\sum_{k=0}^{n-1}\frac{y}{n}\right)=\sum_{n\geq 1}y\log n.
\end{align*}
If $x$ is non-integer, I don’t know much about the cases of $y\in\mathbb{R}$ or $y=0$, or $y_n=\frac{y_1}{n}$ for non-integer $x$ and $y_1\in\mathbb{R}$. However, I can at least give a nice characterization when $y_n$ is real non-negative and non-increasing, and I don’t think this characterization is known.
For $x\geq 2$, define $G(x)=\sum_{n\ge
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