Riemann Zeta Function:
$$\frac{1}{1^{s}}+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\frac{1}{5^{s}}+\frac{1}{6^{s}}+\frac{1}{7^{s}}+\frac{1}{8^{s}}+\frac{1}{9^{s}}+\frac{1}{10^{s}}+\dots$$
"Reverse $2$" Riemann Zeta Function:
$$\frac{1}{1^{s}}+\color{red}{\frac{1}{\sqrt2^{s}}}+\frac{1}{3^{s}}+\color{red}{\frac{1}{2^{s}}}+\frac{1}{5^{s}}+\color{red}{\frac{1}{\sqrt{18}^{s}}}+\frac{1}{7^{s}}+\color{red}{\frac{1}{\sqrt{8}^{s}}}+\frac{1}{9^{s}}+\color{red}{\frac{1}{\sqrt{50}^{s}}}+\dots$$
"Reverse $3$" Riemann Zeta Function:
$$\frac{1}{1^{s}}+\frac{1}{2^{s}}+\color{blue}{\frac{1}{\sqrt3^{s}}}+\frac{1}{4^{s}}+\color{blue}{\frac{1}{\sqrt{35}^{s}}}+\color{blue}{\frac{1}{\sqrt{12}^{s}}}+\color{blue}{\frac{1}{\sqrt{35}^{s}}}+\frac{1}{8^{s}}+\color{blue}{\frac{1}{3^{s}}}+\frac{1}{10^{s}}+\dots$$
"Reverse $10$" Riemann Zeta Function:
$$\frac{1}{1^{s}}+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\frac{1}{5^{s}}+\frac{1}{6^{s}}+\frac{1}{7^{s}}+\frac{1}{8^{s}}+\frac{1}{9^{s}}+\color{green}{\frac{1}{\sqrt{10}^{s}}}+\dots$$
In general, a Reverse $x$ Riemann zeta function takes $n=1,2,3\dots$ and multiplies it with its reverse value (reverse digits) in number base $x$, then takes the square root of the product. After that, raise it to the power of $-s$ as you would for zeta(s).
As you can see, if $n$ is a palindrome in number base $x$, then it remains $n$. Otherwise, it becomes the $\sqrt{n\times\overline{n}}$ where $\overline{n}$ is the number $n$, but its digits are reversed when written in number base $x$.
The value of "Reverse Zeta $x$" of $s$ tends to $\zeta{(s)}$ as $x$ grows bigger. Suppose we take the limit $x\to\infty$, then we have a normal zeta function. It's because that makes the number base a base where each number is a single symbol (one digit), meaning all $n$s reverse back to $n$.
Does something on this already exist somewhere or is this the first time this is mentioned?
How can we check when this will converge for a given $x$?
I suppose the bound slightly varies based on $x$ and converges to that the real part of $s$ must be $\gt1$ as $x$ tends to $\infty$?Can one find closed forms for some $x$ and some $s$?
For example, what is the (is there a) closed form of this when $x=2$ and $s=2$ ?
Can we express the "reverse $x$ zeta function" in the terms of the zeta function?
Also, I couldn't find anything on the reciprocal sums of palindromic numbers.