Let $Y:=[0,1]$ and $X:=[0,t]$ and consider the following curves embedded in $Y^{d+1}:$
$$ \zeta_1( t)=\bigg(\lim_{k \to\infty}\bigg( \int_{Y}\sum_{n=1}^k\varphi_n(x)dx\bigg)^{-1}\int_{X} \sum_{n=1}^k \varphi_n(x)dx,t \bigg) $$
$$ \zeta_2(t)=\bigg(\lim_{k \to\infty}\bigg( \int_{Y}\sum_{n=1}^k\varphi_n(x)dx\bigg)^{-1}\int_{X} \sum_{n=1}^k \varphi_n(x)dx,t,t \bigg) $$
$$... $$
$$ \zeta_d(t)=\bigg(\lim_{k \to\infty}\bigg( \int_{Y}\sum_{n=1}^k\varphi_n(x)dx\bigg)^{-1}\int_{X} \sum_{n=1}^k \varphi_n(x)dx,t, \cdot\cdot\cdot,t \bigg) $$
Here $\varphi_n(x)=\exp\bigg( \frac{\log n}{\log x} \bigg) $. Each of the $\zeta_1(t),\zeta_2(t),\cdot\cdot\cdot, \zeta_d(t)$ are understood to be generators together with the operation of rotation about the long diagonal of the hypercube $(t,\cdot\cdot\cdot,t)$. For example $\zeta_1(t)$ is reflected about $(t,t)$, $\zeta_2(t)$ is rotated about $(t,t,t)$ etc. to obtain hypersurfaces of revolution.
Considering a low dim. example, $Y^{2+1}$, we can generate coordinate projections of $\Psi$ onto its faces. By definition these objects are linear transformations of the structure $\Phi(x)=\sum_{n=1}^{\infty} \varphi_n(x_1)$ and the transformations map that structure onto $Y^{1+1}$ where we gain compactness.
$$\Phi(x)=\sum_{n=1}^{\infty} \varphi_n(x)$$
can be analytically continued to a meromorphic function and is equivalent to a composition with the Riemann zeta function. I'm trying to understand the analogue after $\Phi$ is generalized, but pre-analytic continuation.
For dimension $Y^{1+1},$ the generator $\zeta_1(t)$ can be given explicitly. I also know that the generators are related through a sequence of projections which is pretty clear from the definition I gave. So I know that the hypersurfaces are related to one another, I am just not sure exactly how certain properties extend with this set of generalized surfaces.
For example we know that $\zeta_1(t)$ is a linear transformation of a composition with the Riemann zeta function, therefore we have a linear transformation of the corresponding Euler product at hand. We know that $\zeta_1(t)$ is a lower dimensional version of the surface upstairs as well.
My question is:
Is the surface of revolution associated to the generator, $\zeta_2(t),$ related to the primes/Riemann zeta function? By extension is the hypersurface of revolution associated to the generator $\zeta_d(t)$ related to the primes/Riemann zeta function?
The extensions I've done are all pre-analytic continuation. As I said you could very well directly extend $\Phi(x)$ meromorphically but I want to generalize before this analytic continuation is done, and only after this, look at possible extensions to complex space.
I also think that the surface of revolution for the generator $\zeta_2(t)$ is a double cover for the base space. In other words projection of all the meridians onto $Y^2$ gives $\zeta_1(t)$ unioned with $\zeta_1(t)$ for finite $k$. In some sense I think this would allow to interpolate the base space for non integer $k$.