
The Riemann Hypothesis, explained - seycombi
https://medium.com/@JorgenVeisdal/the-riemann-hypothesis-explained-fa01c1f75d3f#.9hp0wrml3
======
tzs
There are some interesting statements that are equivalent to the Riemann
Hypothesis. What "equivalent" means is that if the statement is true then RH
must be true, and if RH is true then the statement must be true.

Here's one I find particularly nice.

Let s(n) = the sum of the divisors of n, for a positive integer n. For example
s(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.

Let H(n) = 1 + 1/2 + 1/3 + ... + 1/n.

The RH is equivalent to the claim that for every integer n >= 1:

s(n) <= H(n) + exp(H(n)) log(H(n))

This is due to Jeffry C. Lagarias. Here's his paper showing the equivalence:
[https://arxiv.org/abs/math/0008177](https://arxiv.org/abs/math/0008177)

~~~
rfurmani
Here's another one. You'll notice that typically there's slightly more numbers
that factor into a product of an odd number of primes (i.e.
2,3,5,7,8,11,12,13,...) than into an even number of primes
(4,6,9,10,14,15,...). If this is the case then RH is true. You can make this
into a more precise question about the partial sums of the Liouville function
and make it an equivalence.

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vq
3Blue1Brown recently did a video on the Riemann zeta function:
[https://www.youtube.com/watch?v=sD0NjbwqlYw](https://www.youtube.com/watch?v=sD0NjbwqlYw)

It's not as in depth but it has some helpful visualisations that I've never
seen elsewhere.

~~~
Cyph0n
Thank you, that was an amazing video. I literally hit the "Subscribe" button 2
minutes in!

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williamstein
Shameless promotion: I published a book recently trying to explain RH, and my
coauthor gave a great talk about it, which is here:
[http://wstein.org/rh](http://wstein.org/rh)

~~~
seycombi
Can I ask how many times a month (as a mathematics professor) you get 'THE
solution' in your inbox?

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rfurmani
Great article!

One of the things about the Riemann Hypothesis is that the search space for a
proof is more wide than it is deep. For each given idea or approach it doesn't
take relatively long to get to the forefront of what is known, and an expert
can often tell you right from the beginning that the whole class of approaches
may not work due to some known phenomena, or that they would have to involve
certain complications to not pick up on various almost-counterexamples.
Furthermore, in these 150 years not only has the right path/approach not been
found, but there isn't a truly compelling reason to believe that RH is true,
aside from numerical evidence and a belief in beauty.

I think it'd be fascinating to put together an online resource to organize the
possible approaches, list the knowledge prerequisites, show the potential
counterexamples and stumbling blocks to each approach. This would help anyone
interested in the problem, and once it is sufficiently developed it would
allow non-specialists to contribute productively. This could be LaTeX on
github, it could be more of a traditional wiki, but now I really want to get
this going.

~~~
rkowalick
There was actually an entire conference in 1996 pretty much dedicated to talks
about "How _not_ to prove the Riemann Hypothesis" to prevent people from
wasting time on approaches that were known not to work:

[http://www.aimath.org/research/conferences/rh-
conf.html](http://www.aimath.org/research/conferences/rh-conf.html)

~~~
rfurmani
Looks like a great conference, regrettably before my time (and unfortunately
no videos of course). They look like fairly standard topics about the zeta
function, where does "how not to prove RH" come in?

~~~
qubex
I'm not trying to be rude here, but academics in traditionally textual
disciplines that lament lack of video media leave me totally agog.

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ot
This is a great introduction to analytic number theory.

I was expecting the standard dumbed down piece with no mathematical insight
and far-fetched analogies. Instead, the post is mathematically rigorous and
deep, and yet it manages to be completely self-contained and clear. Amazing
work.

~~~
ianai
I bookmarked it for future reference study "if I ever get to it"

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hiq
[nitpicking]

This version of Euclid's proof does not look like the one on Wikipedia, which
is not a proof by contradiction. The article[0] mentions that:

"Euclid is often erroneously reported to have proved this result by
contradiction, beginning with the assumption that the finite set initially
considered contains all prime numbers, or that it contains precisely the n
smallest primes, rather than any arbitrary finite set of primes."

[/nitpicking]

[0]:
[https://en.wikipedia.org/wiki/Euclid's_theorem#Euclid.27s_pr...](https://en.wikipedia.org/wiki/Euclid's_theorem#Euclid.27s_proof)

~~~
paulddraper
Nitpicking indeed.

Euclid's proof: Every finite set of primes {p1,p2,...pn} has a prime, namely
p1p2...pn + 1, not in the set.

Contradiction proof: The hypothetical finite set of all primes {p1,p2,...pn}
would have a prime, namely p1p2...pn + 1, not in the set.

I'd hardly call the latter a different proof.

~~~
tprice7
You made an error: p1p2...pn + 1 is not necessarily prime, but none of its
prime factors are in {p1, ..., pn}. For example, 15 = 2 * 7 + 1 is not prime,
but neither 3 nor 5 are in {2, 7}. I agree with your overall point though.

~~~
daveguy
All primes. Not just the primes you choose. The set of the first n primes here
would be {2,3,5,7}. The product plus 1 is 2 _3_ 5*7+1 = 211 is prime.

~~~
pmiller2
The product of the first n primes + 1 is not always prime, though. See
[https://en.m.wikipedia.org/wiki/Euclid_number](https://en.m.wikipedia.org/wiki/Euclid_number)

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jordigh
Hm, except for using Gamma(n) = (n-1)! instead of Pi(n) = n!, this
presentation seems to follow the notation of this book:

[https://books.google.ca/books/about/Riemann_s_Zeta_Function....](https://books.google.ca/books/about/Riemann_s_Zeta_Function.html?id=ruVmGFPwNhQC)

This is most obvious in the naming of the J function near the bottom, which I
have not seen anywhere else. Riemann originally called that function a generic
f.

At any rate, Edwards' book is great because it develops the theory from a
historical viewpoint. It begins with a very well-annotated exposition of
Riemann's original paper and the rest of the book goes on to explain other
mathematicians' successive results in filling in all of the gaps that Riemann
left in his paper. I recommend this book to any serious student of zeta.

~~~
jorgenveisdal
Books used:

1\. Men of Mathematics (E.T. Bell, 1937)

2\. The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike
(Peter Borwein et.al, 2007)

3\. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in
Mathematics (John Derbyshire, 2004)

4\. Unknown Quantity: A Real and Imaginary History of Algebra (John
Derbyshire, 2007)

5\. Journey through Genius: The Great Theorems of Mathematics (William Dunham,
1991)

6\. Riemann’s Zeta Function (H. M. Edwards, 1971)

7\. Gamma: Exploring Euler’s Constant (Julian Havil, 2009)

8\. The Man Who Loved Only Numbers (Paul Hoffman, 1999)

9\. The Riemann Zeta-Function: Theory and Applications. (Alexander Ivic, 2004)

10\. e: The Story of a Number (Eli Maor, 2009)

11\. An Imaginary Tale: The Story of √-1 (Paul J. Nahin; 1998)

12\. Dr. Euler’s Fabulous Formula (Paul J. Nahin; 2006)

13\. Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of
Prime Numbers (Dan Rockmore, 2006)

14\. Infinity and the Mind (Rudy Rucker, 2004)

15\. The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics
(Karl Sabbagh, 2004)

16\. The Prime Numbers and Their Distribution. (Tenenbaum and Mendès, 2000)

~~~
anaerobia
There is a video
[http://library.fora.tv/2014/04/25/Riemann_Hypothesis_The_Mil...](http://library.fora.tv/2014/04/25/Riemann_Hypothesis_The_Million_Dollar_Challenge),
and the book [https://www.amazon.com/Prime-Numbers-Riemann-Hypothesis-
Barr...](https://www.amazon.com/Prime-Numbers-Riemann-Hypothesis-
Barry/dp/1107499437)

------
seycombi
For those who did not read it or missed it, at the end of the article there is
a link to Jørgen Veisdal's 2013 undergraduate thesis paper.

[http://www.jorgenveisdal.com/files/jorgenveisdal-
thesis13.pd...](http://www.jorgenveisdal.com/files/jorgenveisdal-thesis13.pdf)

------
bloodred92
I'm pretty awful at math and I found this explanation to be wonderful despite
many parts of it going over my head. My brain naturally despises numbers
(dyslexia)but your textual descriptions of formulas seemed to help me bridge
the gap better than most texts.

~~~
mirceal
Math is super interesting. The reason why most people don't like it and/or
think it's hard is because they did not have a great teacher. Once you
understand the notation and grasp the basics it's awesome.

So, it's more the language of math that makes it hard to understand than it is
about the math itself.

~~~
source99
which is really quite sad.

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mrcactu5
here is a more technical discussion by Paul Bourgade @ NYU. I still re-read it
from time to time for orientation.

Quantum chaos, random matrix theory, and the Riemann ζ function

[http://www.cims.nyu.edu/~bourgade/papers/PoincareSeminar.pdf](http://www.cims.nyu.edu/~bourgade/papers/PoincareSeminar.pdf)

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throwaway7645
I went to a talk on the Reimann hypothesis with a physicist friend. The chair
of our math department gave the talk. At one point he stated "Do you know why
the Reimann hypothesis is important?, it's because it's the Reimann
hypothesis". Suffice to say the entire lecture was waaaay over our heads.

------
tzs
> The gamma function Γ(z) is defined for all complex values of z larger than
> zero

What does "larger than zero" mean for complex numbers?

~~~
delhanty
From your other answer it sounds like that you know what the OP means? I'll
bite anyway ...

The Wikipedia article on the gamma function

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

states that "The gamma function is defined for all complex numbers except the
non-positive integers. For complex numbers with a positive real part, it is
defined via a convergent improper integral ..."

So maybe that is what the OP means: the gamma function Γ(z) can be defined by
via convergent improper integral for all complex numbers of z with _real part_
greater than zero.

?

Wikipedia then points out that:

"This integral function is extended by analytic continuation to all complex
numbers except the non-positive integers (where the function has simple
poles), yielding the meromorphic function we call the gamma function."

~~~
tzs
> From your other answer it sounds like that you know what the OP means?

No, I had no idea what he meant. His statement about complex z larger than 0
came right after he gave a plot of gamma for real z from -6 to 6.

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Bahamut
For the curious, much of this is covered in depth in a graduate analytic
number theory first course.

As fascinating as the Riemann Zeta Hypothesis is, I think something almost as
fascinating is the works of Ramanujan - to this day, the genius Indian's work
is studied in earnest, and amongst the most famous number theory work to this
day.

------
ifoundthetao
For a nice historical background on this, you can read "Prime Obsession",
which is a pretty fascinating read.

------
jonaf
This article is about math.

In my ignorance, I mistook the article to be about the monitoring project[1]
-- I expected some kind of proof/theorem behind the monitoring project.

This article is not about the monitoring project.

[1]: [http://riemann.io/](http://riemann.io/)

~~~
Smaug123
In fairness, the Riemann hypothesis is one of the most famous mathematical
open problems, alongside the twin prime conjecture and P vs NP.

------
jonbaer
Great post, I have been reading [https://www.amazon.com/Riemann-Hypothesis-
Greatest-Unsolved-...](https://www.amazon.com/Riemann-Hypothesis-Greatest-
Unsolved-Mathematics/dp/0374529353) for over 10 years now ;-)

------
ianai
I really appreciated the depth here.

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Dzugaru
> The real valued zeta function is given for r and n, two real numbers

followed by a function of one real variable with an infinite sum of
expressions containing integer r.

Sorry, but little things like these completely put me off in math articles.

~~~
bubaflub
Good catch! I read that paragraph the same way and I believe it is a typo.
I've left a comment on the medium article asking the author to clarify that
section.

If you don't mind me asking, what is so off-putting about these types of
mistakes? I'd like to do more personal and professional technical writing
similar to this but I find it a bit daunting, especially when I consider that
my audience could know much more about a subject than I do. Do you have any
suggestions on how I could or should write articles that may contain errors so
I don't upset readers?

~~~
Manishearth
I came across that mistake (and it was pretty obviously a mistake to me, but
that's because I already know quite a bit about the RH), and later was
stonewalled by a different mistake -- the "Plot of the real and imaginary part
of the Riemann zeta function ζ(s) in the interval -5 < Re < 2, 0 < Im < 60"
plot is mislabeled; it's actually a contour graph of the condition "ζ(s) = 0"
on the complex plane with real and imaginary parts plotted separately.

The problem for me is that my way of reading math stuff is to assume that
whatever is said is being true and fit it into a growing mental model as I
read. Many other students I know do that. Once you get into the flow of it
this can get pretty fast without losing out on comprehension. When you come
across mistakes, it leads to a speed bump, and you have to stop and re-collect
your thoughts after this. This only happens to me for math and physics, not
other kinds of technical writing, but it happens.

This doesn't turn me off an article, it's just super annoying.

Typos and stuff are fine, it's the "plausibly true" stuff that gets you. I
suggest spending more time proofreading but not worrying more about it.

------
tokenadult
I like his wording of Euclid's classic proof of the infinity of prime numbers.
That's good exposition to start off the article.

~~~
jorgenveisdal
Thank you! One of my favorites

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ianamartin
___META_ __

Why are people still posting things on Medium?

It's a platform that the founder knows has absolutely no clue about how to
sustain it. After 5 years, and over hundred million dollars of investments.

 __ _end meta_ __

This is a good post, and it should be put somewhere that will keep it.

That said, "RH" stuff now:

Fuck Riemann. An entire school of music theory that has almost nothing to do
with him is named after him. It's a ginormoulsy idiotic way to understand
music.

The people who think of him as a great thinker as it applies to music are
completely out of their minds.

/trolling, sort of. a little.

~~~
isotropy
Bernhard is the number theorist - you are thinking of Hugo.

~~~
ianamartin
You are correct, and I was mistaken.

Rant redacted.

