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Here is the plot of a function $f(x)$ such that: $$ f(x) = \frac {P_ {\lfloor 2 x \rfloor}} {P_ {\lfloor 2 x \rfloor - 1}} $$ where $P_k$ is the $k^{\mathrm{th}}$ Prime Number for x in range [1,300].

Plot Primes

The curves are in the form $1 + \frac {1} {a x^b}$, with $a,b\in\mathbb {R}$ and where I chose arbitrarily $a$ and $b$, $b\simeq 1.22$ in the image above.
Personally, I was very surprised whith what seems very regular shapes occurring again and again along the curves.
Can someone explain this?

I deleted the first post about this, because it seems more like a Mathematica bug than something else, but I cannot find what's wrong. Here is the Mathematica code:

Block[{x},
Module[{ a, b, c, xa, xb, plotall, plots0, plots1},
a = Prime[Floor[2*x - 1]];
b = Prime[Floor[2*x]];
xa = 1; xb = 300; c = 1.22;
plots0 = { b/a};
plots1 = {1 + 1/2.21/x^c, 1 + 1/1.1/x^c, 1 + 1/0.725/x^c, 
1 + 1/0.435/x^c, 1 + 1/0.31/x^c, 1 + 1/0.195/x^c, 
1 + 1/0.126/x^c  };
plotall = Join @@ {plots0, plots1};
Panel@Plot[  Tooltip@plotall, {x, xa, xb} , AxesLabel -> {x, y}]
]
]

As also suggested, here is the plot of: $$ f (x) = \frac {P_ {\lfloor x \rfloor}} {P_ {\lfloor x \rfloor - 1}} $$

enter image description here

and the interactive Mathematica code:

Block[{x},
Manipulate[
Module[{a, b, xa, xb, plotall, plots0, plots1},
a = Prime[Floor[x] - 1];
b = Prime[Floor[x]];
xa = 2; xb = 300;
plots0 = { b/a};
plots1 = { 1 + 1/r1/x^c1 };
plotall = Join @@ {plots0, plots1};
Plot[ Tooltip@plotall, {x, xa, xb} , AxesLabel -> {x, y}]
]
, Style["Global Options", Bold, "Panel"]
, {{r1, 0.83}, 0, 3, Appearance -> "Open"}
, {{c1, 1.25}, 0, 5, Appearance -> "Open"}
]
]

Since Julian Aguirre's answer, I just wanted to add the following graph, where the curves are in the form $1+ \frac {7} {4}.\frac {n} {x\text { Log} (x)}$ for $n$ integer in range[1,10]...

enter image description here

Eddy Khemiri
  • 533
  • @Clayton, the nth prime number is given by the floor functions $\lfloor 2 x \rfloor$ and $\lfloor 2 x \rfloor - 1$ in the formula. – Eddy Khemiri Oct 30 '13 at 12:38
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    I think it has a lot to do with the fact that the distance between any two primes is about ln$(p)$ and the size of the primes is getting larger at a much faster rate. This fact would cause your graph to approach 1 because you are essentially taking the ratio of 2 very close numbers. That explains why your graph looks a lot like $\frac{1}{ln(p)}$. – Rocket Man Oct 30 '13 at 12:42
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    Why not just graph $\frac{P_{\lfloor x\rfloor}}{P_{\lfloor x\rfloor}-1}$? The $2$ is just a scaling factor that adds noise to the function. – Thomas Andrews Oct 30 '13 at 12:43

1 Answers1

2

What you see is due to the gaps between prime numbers. $$ \frac{P_{2k}}{P_{2k-1}}=1+\frac{P_{2k}-P_{2k-1}}{P_{2k-1}}. $$ The curves you see are formed by primes with gaps $2$, $4$, $6$, \dots

¿What can we say about the curves? Using the Prime Number Theorem, the primes with gap $g$, where $g$ is a positive even number, lie close to the prph of the function $$ f_g(x)=1+\frac{g}{(2\,x-1)\ln(2\,x-1)}. $$ You can see it in the following picture, where points are colores according to the gap: enter image description here

Julián Aguirre
  • 76,354