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- Feb 13, 2012

- 1,704

The prime number theorem is related to the

$\displaystyle \pi(x) \sim \frac{x}{\ln x}\ (1)$

In the last two centuries great work has be done about this problem and one of the most remarkable result is the

$\displaystyle \lim_{ x \rightarrow \infty} \frac{\pi(x)}{\frac{x}{\ln x}} = 1\ (2)$

Kind regards

$\chi$ $\sigma$

- Mar 22, 2013

- 573

\(\displaystyle \frac{Ax}{\log x} < \pi(x) < \frac{Bx}{\log x}\)

It was later established by Hadamard and de la Vallée Poussin the one shown by chisigma in the previous post.

PS I think there is nothing in the proof of PNT that cannot be understandable to an 18 year onld.

- Feb 13, 2012

- 1,704

It depends from how many 'neurons' the young boy has... for example the fellow in the figure below, when hi was thirteen, was capable to reproduce the Allegri's Miserere, rigorously taken as a 'copyright' from Vatican, after having heard it onle one time...... I think there is nothing in the proof of PNT that cannot be understandable to an 18 year old...

Kind regards

$\chi$ $\sigma$

I highly recommend the book if you haven't read it, by the way.

- Feb 13, 2012

- 1,704

I wonder why a theorem that has been rigolusly demonstraded more that hundred years ago by Jaques Hadamard and Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin needs today of 'numerical validations'... it is better for us to spend our time to perform more mathematical advances... or not?...

I highly recommend the book if you haven't read it, by the way.

Kind regards

$\chi$ $\sigma$

- Mar 22, 2013

- 573

Actually \(\displaystyle x/log(x)\) is nothing more than a match of the order of magnitude. A far better (unconditional) approximation is the one that uses that zeros of zeta which is impressive if illustrated neatly (although I don't think any exists since evaluation of Li at complex values and manipulating that much zeros of zeta is very tiresome).akward said:n "Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics", John Derbyshire introduces the prime number theorem by displaying a table of N, ln(N), and N / pi(N), where pi(N) is the number of primes less than or equal to N, for some large values of N, ranging from 10^3 to 10^18. The numerical evidence that ln(N) is close to N / pi(N) is then strong. Maybe you could do something similar.

I highly recommend the book if you haven't read it, by the way.

Balarka

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- Aug 18, 2013

- 76

Says the 8th grade graduate!I think there is nothing in the proof of PNT that cannot be understandable to an 18 year old.

From the OP:I wonder why a theorem that has been rigolusly demonstraded more that hundred years ago by Jaques Hadamard and Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin needs today of 'numerical validations'... it is better for us to spend our time to perform more mathematical advances... or not?...

Kind regards

$\chi$ $\sigma$

"I am not seeking a proof of the result but something which will have an impact and motivate a student to study mathematics in the future."