A curious result

\begin{equation}

\lim_{x\to +\infty} \frac{\Pi(x) \ln(x)}{x}=1

\end{equation}

where \(\Pi(x)\) is the number of primes not greater than \(x\).

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# Author: Jorge Sofo

## How many primes…

## Wonderful collection of 2D curves, 3D curves, surfaces, fractals, and polyhedra

## Three rules for a scientific career

## The coolest functions

## Browein integral

## The Pythagorean Cup, created by Phythagoras of Samos

## 3Blue1Brown: Solving 2D equations using color, a story of winding numbers and composition

## Jorge Sofo Interview by Oficina de Prensa of Instituto Balseiro (in spanish)

## Osmium

## A Positive Reminder by J. A. Lindon

### Footnotes

Department of Physics and Materials Research Institute

Jorge Sofo is a Professor of Physics at Penn State.

A curious result

\begin{equation}

\lim_{x\to +\infty} \frac{\Pi(x) \ln(x)}{x}=1

\end{equation}

where \(\Pi(x)\) is the number of primes not greater than \(x\).

Can be found here.

- Don’t write a paper

… you wouldn’t read yourself. - Don’t work with anyone

… you don’t respect and admire. - Work only with people

… you enjoy and trust.

(Posted on the door of Ale Valdivia, Universidad de Chile, May 2019.)

These elements have the highest density:

1. Osmium, 22.6 g/cc

2. Iridium, 22.4 g/cc

3. Platinum, 21.45 g/cc

4. Rhenium, 21.2 g/cc

5. Uranium, 20.2 g/cc

“A carpenter named Charlie Bratticks,

Who had a taste for mathematics,

One summer Tuesday, just for fun,

Made a wooden cube side minus one.

Though this to you may well seem wrong,

He made it minus one foot long,

Which meant (I hope your brains aren’t frothing)

Its length was one foot less than nothing,

Its width the same (you’re not asleep?)

And likewise minus one foot deep;

Giving, when multiplied (be solemn!),

Minus one cubic foot of volume.

With sweating brow this cube he sawed

Through areas of solid board;

For though each cut had minus length,

Minus times minus sapped his strength.

A second cube he made, but thus:

This time each one-foot length was plus:

Meaning of course that here one put

For volume, plus one cubic foot.

So now he had, just for his sins,

Two cubes as like as deviant twins:

And feeling one should know the worst,

He placed the second in the first.

One plus, one minus — there’s no doubt

The edges simply canceled out;

So did the volume, nothing gained;

Only the surfaces remained.

Well may you open wide your eyes,

For those were now of double size,

On something which, thanks to his skill,

Took up no room and measured nil.

From solid ebony he’d cut

These bulky cubic objects, but

All that remained was now a thin

Black sharply-angled sort of skin

Of twelve square feet — which though not small,

Weighed nothing, filled no space at all.

It stands there yet on Charlie’s floor;

He can’t think what to use it for!”

This poem by J. A. Lindon[1] was popularized by the inimitable Martin Gardner in “The Magic and Mystery of Numbers[2]”.

Thank you Alon Amit in Quora

[2] https://books.google.com/books?i…