
After Prime Proof, an Unlikely Star Rises - digital55
https://www.quantamagazine.org/20150402-prime-proof-zhang-interview/
======
dude_abides
A writeup by his doctoral advisor about his memories of Yitang Zhang:
[http://www.math.purdue.edu/~ttm/ZhangYt.pdf](http://www.math.purdue.edu/~ttm/ZhangYt.pdf)

~~~
logicallee
I skimmed it, but in the intro why does he say:

    
    
      "The most natural conjecture is that the prime numbers appear randomly..
      If the twin prime conjecture is correct, then we may conclude that the
      [primes] are not constructed randomly" 
    

why does he say a random distribution of primes would imply the twin prime
conjecture is false?

My intuition is the opposite: a random distribution of primes implies the twin
prime conjecture is true. Because suppose there are finite twin primes, set
them aside; then test pairs of numbers above the largest of them. Since we
assume the twin prime conjecture is false, the chances that random pairs above
that number are both prime is exacty 0; but that would mean the primes are
_not_ distributed randomly...

~~~
ilyagr
I thought about this for a second, and it seems that you are right: if the
primes were distributed randomly, the twim prime conjecture would be true (see
below). Are there any analytic number theorists in the audience to explain
what the quote in the parent post is supposed to mean and/or fix my mistakes?

The prime number theorem [1] states that the probability that a natural number
N is prime is approximately 1/log(N). So, the probability that N & N+2 are
both prime is about 1/log(N)^2. If they were distributed randomly, there would
be about

integral(1/log(x)^2 dx) from x=2 to x=N

twin primes between 1 and N. This integral converges to infinity as N grows
large[2] (wish I remembered enough caclulus to do it by hand). So, there is
some N for which you'd expect to have 10^100 twin primes less than N, etc.

So it seems that if you decided by a coin toss whether a number should be
prime (where you'd make it prime with probability 1/log(N)), there would be
infinitely many twin primes.

Wikipedia seems to confirm it's a reasonable estimate [3].

 _UPDATE:_ By the way, the same argument implies that if primeness was decided
by a coin toss, there would be infinitely many N such that N and N+1 are
prime. There is, however, only one such N (namely 2). So, primeness is not
decided by a coin toss, and the above is not a proof of the twin primes
conjecture :).

[1]:
[http://en.wikipedia.org/wiki/Prime_number_theorem](http://en.wikipedia.org/wiki/Prime_number_theorem)

[2]:
[http://www.wolframalpha.com/input/?i=Integrate[1%2FLog[x]^2%...](http://www.wolframalpha.com/input/?i=Integrate\[1%2FLog\[x\]^2%2C+{x%2C+2%2C+Infinity})]

[3]:
[http://en.wikipedia.org/wiki/Twin_prime#First_Hardy.E2.80.93...](http://en.wikipedia.org/wiki/Twin_prime#First_Hardy.E2.80.93Littlewood_conjecture)

~~~
ilyagr
Another UPDATE:

I a cousin comment, Retric proves that the integral of 1/log(x)^2 diverges.
The point is that log(n)^2 < n for x > 2 (you can prove this by calculus: the
only maximum of the function log(x)^2/x is at x=e). So, 1/log(x)^2 > 1/x, and
the integral of 1/x diverges.

The most fun way to prove the last statement is that the integral of 1/x from
1 to infinity is greater than 1/2 + 1/3 + 1/4 + 1/5 + ... > 1/2 + 1/4 + 1/4 +
1/8 + ... = 1/2 + 2 * 1/4 + 4 * 1/8 + ... = 1/2 + 1/2 + 1/2 + ....

------
bjwbell
Yitang Zhang brings up the deep question of why does someone like him suffer
in the world today. He is not showey at all but quiet and unassuming. In
particular he is not un-confident but he's not showey in his confidence.

I suppose we don't value those traits.

~~~
davidp
I think it's more straightforward and less depressing than that. For most
people it's simply hard to tell the difference between someone with those
traits ("still waters run deep"), and someone who just doesn't have much to
offer[0]. We do tend to correct those misperceptions given enough exposure and
investment in the relationship, but it takes time -- and there's only so much
time in a day.

[0] modulo subject X or situation Y -- we all have talents

------
hendzen

      Still in the field of number theory, 
      I may not have only one problem to think about,
    
      but a couple of problems, like the distribution 
      of the zeros of the zeta functions and the L-functions.
    

Interesting to hear that he seems to be working on the Riemann hypothesis.

~~~
nmc
Interesting yet unsurprising. Proving the Riemann hypothesis ( _ie_ all non-
trivial zeros of the zeta function are distributed along the line _x_ = 1/2)
is equivalent to proving Riemann's explicit formula for the prime counting
function, which defines the distribution of prime numbers.

In the end, the Riemann hypothesis is a kind of sacred Graal for anyone
interested in prime numbers.

------
mikek
Is there any way to view the film Counting From Infinity online?

~~~
pronoiac
Probably not; canistream.it comes up empty.

------
madengr
Good to read a story about an underdog.

------
rasz_pl
>Andrew Granville “There’s no way that somebody I’ve never heard of has done
this.”

What an asshole.

------
trafficlight
> “Never heard of him. Absolutely never heard of him,” said Andrew Granville,
> a number theorist at the University of Montreal, in Counting From Infinity.
> When Granville heard about the result and the techniques that Zhang used, he
> recalled saying, “There’s no way that somebody I’ve never heard of has done
> this.”

Fuck that guy.

~~~
whitewhim
When you are an expert in an academic field, you typically have spent many,
many years reading about the problems in the field and the main ways to attack
them. The same authors, will typically pop up again and again. Through
conferences and collaborations one gets to know the main contributors in the
field. On top of this, it is very rare for someone to break into a field with
a major discovery. Normally people tend to hone their skills on more minor
problems over time and obtain a deep understanding of the material. To top all
of this off, academics quite regularly receive emails from crackpots claiming
to have found a solution to p=np, etc.

I am not saying that Dr. Granville was right to immediately dismiss the
concept of an unknown solving such a problem. I am just trying to provide some
background for why he may of reacted as he did.

~~~
fspeech
Hmm there is a difference between a unknown and a crank. Zhang was a trained
Ph.D. mathematician with an academic lecturing position. Now I didn't even
finish my Ph.D. in physics but I know enough about how to do research to not
submit a proof that is not rigorous enough to warrant peer review. One should
not refuse to peer review a unknown researcher with the right
credentials/trainings who make serious arguments in line with academic
practices.

Same as you this is not a comment on Dr. Granville. I wrote in response to
your comment on how the field is typically cloistered among a few insiders. I
don't think that justifies refusing to review an unknown (as in not famous)
researcher who demonstrates training and seriousness (which fortunately they
didn't in Zhang's case but Zhang also was extra careful in making his ideas
crystal clear). This happens too rarely for the insiders to claim undue
burden. Cranks are very obvious to identify (a lack of literature review and
understanding of previous works is a tell-tale sign).

~~~
seanmcdirmid
Cranks and unknowns often look alike. Heck, researcher who diverge just a bit
from accepted doctrines often look like cranks until either they repent or
their approach bears obvious fruits.

~~~
angdis
Hmmm, that is plausible, but I can't think of any modern example where some
deviation has resulted in a qualified researcher being categorized as a
crank-- ignored maybe, or perhaps dismissed but definitely not called a
"crank". Actual cranks are very very easy to spot and people won't bother
wasting effort addressing them.

~~~
seanmcdirmid
I'm not sure. Crank is often used as an ad hominem in science, so it is
difficult to tell mudslinging from real crankiness, it depends on the
sincerity of the one applying the label, and also on the severity of the
crankiness. And many people have trouble with groking more benign
eccentricity.

