# 3D Plot of The Riemann Zeta Function

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!

• 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. Oct 10, 2014 at 12:35
• 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. Oct 11, 2014 at 1:35

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]


• @All please Upvote +1000k toonight for kguler. Oct 22, 2014 at 21:59
• @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?
– kglr
Oct 22, 2014 at 22:05
• If i publish an article with, how can add your credit records? Best. Oct 22, 2014 at 22:11
• 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`?
– kglr
Oct 22, 2014 at 22:34