Zeta function and primes number theorem

[Liu]

Riemann Hypothesis states that:

The zeros of $latex \zeta (s)$ in the critical strip: $latex 0\leq Re
(s) \leq 1$ lie on the line $latex Re(s)=1/2$.

Some of the greatest minds have tried unsuccessfully to prove the
notion. It is said that famous and beautiful mind of John Nash is
included. And this hypothesis is more or less related to his mental
problems later in his life.

The fact that interests me the most is if you search for ‘Riemann
Hypothesis’ at arXiv.org, you will find tons of ‘proofs’ and
‘counterexamples’ preprints. I guess most of these proofs, if not all
of them, must contain errors. What is really important in this is
that, in a way, this phenomenon shows how a great mathematical
conjecture influences the development of this field.

The simple corollary of Riemann hypothesis that the zeta function has
no zeros on the line $latex Re(s)=1$ is actually equivalent with the
prime number theorem: that $latex \pi(x)$, denoting the number of
primes less than or equal to x, goes to infinity as the speed of
$latex \frac{x}{\log x}$. The whole Chapter 7 of Stein’s book complex
analysis is dealing with this theorem.

One notices the fact that the sum of all the reciprocals of all
primes, $latex \sum_{n} \frac{1}{p}$, diverges. I have read several
great proofs of this well-known theorem, about which I links to the
wiki.

http://en.wikipedia.org/wiki/Proof_that_the_sum_of_the_reciprocals_of_the_primes_diverges

[Akhil Mathew]

Regarding the prime number theorem, there is a very nice article by
Zagier in the Monthly that gives a proof of the prime number theorem
with no prerequisites other than basic number theory and complex
analysis in three or four pages.

(Citation: The American Mathematical Monthly, Vol. 104, No. 8. (Oct.,
1997), pp. 705-708.)

Soprounov has written up an exposition of the modifications necessary
for the prime number theorem in arithmetic progressions, at
http://www.math.umass.edu/~isoprou/pdf/primes.pdf.

Robert Ash has included Newman's proof of PNT (which I think is what
Zagier presents) in his online textbook on complex variables:
http://www.math.uiuc.edu/~r-ash/CV.html. (Incidentally, Ash has many
other interesting textbooks on his website on other subjects.)

> http://en.wikipedia.org/wiki/Proof_that_the_sum_of_the_reciprocals_of...