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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!

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    $\begingroup$ Welcome to Mathematica.Stackexchange! If you've made some headway e.g. managed to integrate some of the expressions for example it would be good to post that in order to show that you have made an effort. $\endgroup$ – C. E. Oct 10 '14 at 12:35
  • $\begingroup$ Echoing @Pickett's statement, you should show us how you are entering these equations, since I suspect that task is harder than Plot3D. I'm worried you might be putting the cart before the horse here. $\endgroup$ – bobthechemist Oct 11 '14 at 1:35
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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]

enter image description here

| improve this answer | |
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  • $\begingroup$ @All please Upvote +1000k toonight for kguler. $\endgroup$ – Pierre Oct 22 '14 at 21:59
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    $\begingroup$ @Pierre, glad this was useful for you. Yes you can use any setting for ColorFunction such as "Rainbow". Not sure what "the cross strip" is? $\endgroup$ – kglr Oct 22 '14 at 22:05
  • $\begingroup$ If i publish an article with, how can add your credit records? Best. $\endgroup$ – Pierre Oct 22 '14 at 22:11
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    $\begingroup$ Pierre, re citation of Mathematica.SE content in published work, there must be something under the link legal. Re the last question, you mean how to add the constraint 'n is Odd`? $\endgroup$ – kglr Oct 22 '14 at 22:34

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