3D Plot of The Riemann Zeta Function

Question

I am a new user and i would appreciate your help to visualize the 3D of the following function.

Edit

I tried this

ListPlot3D[{n, x, z} /. Solve[(n^2 + x^4)/(4x^2) == z && 0 < n < 101 && 0 < x < 101 && 
0 < z < 101111, {n, x}, Integers], InterpolationOrder -> 0, 
Mesh -> None, ColorFunction -> "Red", Filling -> Bottom, BoxRatios -> 1]

Here is the link of the plot on the plane of the Riemann Zeta Function. Also, i need more rep to add a picture. Thank you.

For every odd number on the form $n=pq$, where $p$ and $q$ are primes, there exists $\zeta(s)=0$ such that $$\int^{n}\int_{2}^{\alpha}\left(\bigtriangleup-\bigtriangledown\right)f(n,\alpha)dnd\alpha=\zeta(s)\int^{n}\int_{2}^{\alpha}f(n,\alpha)dnd\alpha,$$ with $$\bigtriangleup=in^{s}\left(\left(2\alpha_{p}\right)^{-s}+\left(2n^{-1}.\alpha_{p}\right)^{s}\right)$$ and $$\bigtriangledown=in^{s}\left(\left(2\alpha_{q}\right)^{-s}+\left(2n^{-1}.\alpha_{q}\right)^{s}\right)$$
where $\alpha\in P$ denotes the primes lower than $n$, $i=\{-1,1\}$ and $s=2.$

Edit

For the plot it's useful to consider that the equivalent should be $$\bigtriangleup=\frac{n^{2}+x^{4}}{4x^{2}}=\bigtriangledown,$$ where $x=p=q=\alpha.$

Then it would be a strong representation of the Riemann critical line. Merci Beaucoup!

Answer 1

Perhaps Reduce may be easier to work with:

soln = Reduce[(n^2 + x^4)/(4 x^2) == z && 0 < n < 101 && 0 < x < 101 &&
     0 < z < 101111, {n, x, z}, Integers] /. Or | And -> List

{{z == 2, n == 4, x == 2}, {z == 5, n == 8, x == 2}, {z == 5, n == 8, x == 4}, {z == 8, n == 16, x == 4}, {z == 10, n == 12, x == 2}, {z == 10, n == 12, x == 6}, ... }

nxz = soln[[All, All, 2]]

{{4, 2, 2}, {8, 2, 5}, {8, 4, 5}, {16, 4, 8}, {12, 2, 10}, {12, 6, 10}, {24, 4, 13}, {24, 6, 13}, {16, 2, 17}, {16, 8, 17}, {36, 6, 18}, {32, 4, 20}, {32, 8, 20}, ...}

ListPlot3D[nxz, InterpolationOrder -> 0, Mesh -> None, 
 ColorFunction -> (Red &), Filling -> Bottom, BoxRatios -> 1]