Consider the step function
$$A(x)=\sum_{n=1}^\infty e^{\mathrm{floor}\bigg(\frac{\log n}{\log x}\bigg)+\mathrm{floor}\bigg(\frac{\log n}{\log (1-x)}\bigg)} = \prod_{\mathrm{ p~ prime}} \frac{1}{1-e^{\mathrm{floor}\bigg(\frac{\log p}{\log x}\bigg)+\mathrm{floor}\bigg(\frac{\log p}{\log (1-x)}\bigg)}} $$
Here is $A(x)$ generalized to one higher dimension. (desmos struggles to render it)
$A(x)$ cannot be studied using analytic techniques clearly, but it does behave in a way like a zeta function because of its description as a product of primes.
What does one make of this "non-analytic" structure that's expressible as primes? Could $A(x)$ be useful in some respect and what tools could be used to study it, if analytic tools are out the window?
Why complicate things with the floor function?
I suggest you study
$$f(s)=\sum\limits_{n=1}^{\infty} e^{\frac{\log(n)}{\log(s)}+\frac{\log(n)}{\log(1-s)}}=\zeta \left(-\frac{1}{\log (s)}-\frac{1}{\log (1-s)}\right),\quad0<\Re(s)<1\tag{1}$$
with functional equation
$$f(s)=f(1-s)$$
which is illustrated in the Figure below.