
A Friendly Introduction to The Riemann Hypothesis [pdf] - signa11
http://www.math.jhu.edu/~wright/RH2.pdf
======
soVeryTired
Most people have heard of the Sieve of Erathosthenes: put all positive
integers in a list, and delete the number one. Now mark two as prime and
delete all multiples of two. Now mark the next remaining number prime and
delete all multiples of that number. Proceed ad infinitum.

There's a randomised counterpart to the Sieve of Erathosthenes called the
Hawkins ransom sieve: Delete the number 1. Mark two as prime, and _delete all
the larger integers with probability 1 /2_. Now mark the smallest remaining
integer (n, say) prime, and delete all the remaining larger integers with
probability 1/n. Proceed ad infinitum.

One of my favourite results in mathematics is that the Riemann hypothesis
(i.e. the error bound for the prime number theorem) holds for the Hawkins
random primes with probability 1. The proof is only about five pages:
[https://eudml.org/doc/89234](https://eudml.org/doc/89234)

~~~
johnloeber
That's very cool; thanks for sharing.

------
maho
An interesting read! Unfortunately the author omits the mindblowing (at least
for me) proof of why the series

Z(s) = 1 + 1/(2^s) + 1/(3^s) + 1/(4^s) + ...

is equal to the infinite product over the terms "1/(1-p^(-s))", with one term
for each prime number p.

This proof does not need a lot of math [1], and if you know the formula for a
geometric series, you can also "prove it" by hand-waving: Each term in the
product is a geometric series, and if you multiply all these series out, the
result is a sum over all possible permutations of prime-factored numbers
(well, the inverses of them). Since each natural number has exactly one such
represnetation, it's effectively a sum over all 1/(n^s).

Of course, you need some rigor to show that you are actually allowed to
rearrange terms of an infinite product of inite sums like that...

[1]:
[https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_for...](https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function#Proof_of_the_Euler_product_formula)

------
robertelder
Long before I went to university, I became obsessed with the Riemann
hypothesis and one of the first things I put on my site was a simple
explanation of the Riemann Hypothesis. This was before I ever actually studied
math formally, so it's not a very authoritative explanation. For whatever
reason, this explanation is now on the first page of search results in
google.ca for "Riemann hypothesis" in Canada, and at the top of page 2 on
google.com. I get lots of random blog posts citing the explanation I wrote,
and every once in a while I get emails from crazy people who think they have
solved it providing me with their non-sensical proofs.

The problem itself is amazing (even though I don't really understand it). The
infinite series is so simple, but it generates so much interesting complexity
when you look at it visually:

[https://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Ze...](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Zeta_polar.svg/1195px-
Zeta_polar.svg.png)

[http://mathworld.wolfram.com/RiemannPrimeCountingFunction.ht...](http://mathworld.wolfram.com/RiemannPrimeCountingFunction.html)

------
btilly
The Riemann hypothesis is equivalent to a number of interesting statements.
One that I like is this.

The Möbius function, defined by
[http://mathworld.wolfram.com/MoebiusFunction.html](http://mathworld.wolfram.com/MoebiusFunction.html),
shows up in various sieving operation. It is 0 for a random integer with
probability (1 - 6/pi^2) and otherwise has apparently even odds of being +-1.

Its sum is called Mertens function, see
[http://mathworld.wolfram.com/MertensFunction.html](http://mathworld.wolfram.com/MertensFunction.html).
The Riemann conjecture is equivalent to saying that the growth of Mertens
function is o(n^(0.5+e)) for every e > 0\. For a long time it was thought to
be bounded above by n^0.5, but this is known to be wrong.

If you replace the Möbius function with a function that is -1, 0 and 1 with
the right probabilities, then with probability 1 it is O((n *
log(log(n))^0.5). Which would prove the Riemann hypothesis. (This intuition
would have kept Mertens from conjecturing that it is bounded by +-sqrt(n)...)

------
shas3
If you are interested in a longer account, I highly recommend
mathematician/popularizer Marcus DuSautoy's Music of the Primes.
[https://en.m.wikipedia.org/wiki/The_Music_of_the_Primes](https://en.m.wikipedia.org/wiki/The_Music_of_the_Primes)

~~~
Aelinsaar
I would add John Derbyshire's 'Prime Obsession' to that, which adds a nice
historical context to the issue.

------
livatlantis
This is fantastic! I know next to nothing about number theory (despite my
interest in primes) and this had me in stitches. Thanks for the link.

------
leblancfg
That article was really a great refresher on something I barely remember from
undergraduate courses. I'm left a little unfazed about the style of prose, but
I guess that's the price you have to pay for dat dankness, yo.

~~~
acqq
Yes, I liked the content _in spite_ of the "trolling," since there is enough
of actual math, the author actually knows the field and it is presented simple
enough to make the math path enjoyable.

I'm sure it would work good even with less or no "trolling." It is possible to
make the approachable text even without the jokes based on misinformation. I
see the author made the book with that in the title, so it was obviously a
selling point for him, but the quite clear approach to the math material (when
he reader manages to ignore the trolling) is actually good. I have nothing
against the funny presentation of the actual biographical facts however.

The misinformation is, however, problematic. The author even states in the
foreword that he "hopes that some of those will appear in Wikipedia."
Unfortunately, it really can happen.

------
tempodox
I'm enjoying this text immensely. Math should be presented this way more
often.

~~~
Kinnard
A new pedagogical paradigm?

~~~
tempodox
I wouldn't necessarily go that far. My personal taste may not be
representative.

------
pjdorrell
Every now and then someone posts some humorous easily-digestible technical
content to Hacker News and commenters say stuff like "All
mathematics/science/software development textbooks should be written this
way", and some commenters throw out a few similar links, and the story slowly
drops down the HN front page, as stories do, and then it's mostly forgotten,
until the next one comes along.

If education matters, and if human intelligence augmentation matters, then we
should be researching how to make difficult technical material more easily
digestible, for example by enhancing it with an appropriate level of humour.

Millions of dollars are being spent on making machines more intelligent (or is
it billions?). Someone should be spending some time, money and effort on
making people smarter.

Also, compared to other possible human intelligence augmentation technologies,
writing textbooks in a more humorous style is less intrusive than taking
"smart pills" or placing electrodes inside your brain.

~~~
dangom
Through the history of mankind we've been researching ways to make difficult
technical material more easily digestible. And may the invention of print
itself, and the subsequent invention of "the textbook" be a showcase example
of this.

Though humorous style is less intrusive than taking "smart pills" or placing
electrodes inside our brains, it is very context-sensitive. I believe humour
should definitely be used more often in oral presentations of complicated
material, where the presenter is able to "feel" the context of those
listening; but maybe not so much in written text.

To make a point, there are many technical textbooks that use funny examples
and exercises as a way to draw students attention - and that does generally
work quite well - but it can be quite a tiring read if you are going through
the material multiple times.

------
iamtheneal
If you're looking for something with a little more detail (but that still
doesn't require a lot of mathematical background), you should check out
[https://github.com/williamstein/rh](https://github.com/williamstein/rh)

------
dajohnson89
The author has a book, _Trolling Euclid_ [0], on Amazon.

[0][https://www.amazon.com/Trolling-Euclid-Irreverent-
Mathematic...](https://www.amazon.com/Trolling-Euclid-Irreverent-Mathematics-
Important/dp/1523466464)

------
schoen
This style of humor reminded me very strongly of Dave Barry.

[https://en.wikipedia.org/wiki/Dave_Barry](https://en.wikipedia.org/wiki/Dave_Barry)

~~~
signa11
james-mickens has a _very_ similar style as well. for example, if you have not
read 'the night watch'
([http://scholar.harvard.edu/files/mickens/files/thenightwatch...](http://scholar.harvard.edu/files/mickens/files/thenightwatch.pdf))
check it out.

you might like it :)

~~~
schoen
That is definitely similar! Thanks for the suggestion.

I absolutely adored Dave Barry's work when I was in middle school (I often
couldn't read a single paragraph aloud without laughing out loud) but I find
it less enjoyable today. Maybe my taste in humor has shifted somewhat.

------
uptownfunk
Someone needs to rewrite Rudin in this way.

------
Kinnard
Can I double upvote!

