B How does the Riemann Hypothesis/Riemann Zeta function even work?

[Beyond3D]

TL;DR Summary

I don't know how the Riemann hypothesis works, need a rundown on the zeta function and how that finds primes

I understand that $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}\cdots$$
I understand what an infinite series is. I know n goes on to infinity and that s is the function input. I just don't know how the heck this finds primes? Can someone explain? I have minimal knowledge of imaginary numbers other than $\sqrt{-1}=i , i^2=1, i^3=-1\cdot i=-i,i^4=-1^2=1$

Highschooler here but this is not HS level so I tagged it as graduate. Please help!

 

[Nugatory]

Highschooler here but this is not HS level so I tagged it as graduate.

But do you want an answer that makes sense at the graduate level, or that makes sense to a high schooler with the math background that you've described in your post? The thread level prefix is for the sort of answer you want.

 

[fresh_42]

I have tried to answer this in that article here:
https://www.physicsforums.com/insights/the-history-and-importance-of-the-riemann-hypothesis/
but I'm unsure whether it is at the level you are looking for.

 

[Beyond3D]

But do you want an answer that makes sense at the graduate level, or that makes sense to a high schooler with the math background that you've described in your post? The thread level prefix is for the sort of answer you want.

I want an answer that makes sense to a highschooler who has knowledge of all highschool mathematical concepts. I wish I was graduate-level smart.

 

[fresh_42]

I want an answer that makes sense to a highschooler who has knowledge of all highschool mathematical concepts. I wish I was graduate-level smart.

Have you tried to read it? The difficulty is that we need at least some notations to answer it, for example
$$
\large{\operatorname{RH}(\theta)\quad \Longleftrightarrow\quad \pi(x)=\operatorname{Li}(x) + O\left(x^{\theta +\varepsilon }\right)\text{ for all }\varepsilon >0}
$$
which basically says that the Riemann hypothesis is true if and only if the function $\pi(x)$ that counts the prime numbers behaves like the integral logarithm $\displaystyle{\int_2^x \dfrac{1}{\log(t)}\,dt}.$ This relates the distribution of primes to the validity of the Riemann hypothesis.

Here is another article that tries to answer your question less technically:
https://web.archive.org/web/2008072...om/news/2006/03/prime_numbers_get_hitched.php

 

[Beyond3D]

Have you tried to read it? The difficulty is that we need at least some notations to answer it, for example
$$
\large{\operatorname{RH}(\theta)\quad \Longleftrightarrow\quad \pi(x)=\operatorname{Li}(x) + O\left(x^{\theta +\varepsilon }\right)\text{ for all }\varepsilon >0}
$$
which basically says that the Riemann hypothesis is true if and only if the function $\pi(x)$ that counts the prime numbers behaves like the integral logarithm $\displaystyle{\int_2^x \dfrac{1}{\log(t)}\,dt}.$ This relates the distribution of primes to the validity of the Riemann hypothesis.

Here is another article that tries to answer your question less technically:
https://web.archive.org/web/2008072...om/news/2006/03/prime_numbers_get_hitched.php

I appreciate the effort, but could you explain it in simpler terms? I really don't have that much math knowledge, that is part of the reason I joined these forums.

 

[phyzguy]

The book "Prime Obsession" by John Derbyshire I think will answer your questions at the level you want. It will take some time to read, but is worth the effort. Here's a link on Amazon, but you can find it elsewhere.

 

[fresh_42]

I appreciate the effort, but could you explain it in simpler terms? I really don't have that much math knowledge, that is part of the reason I joined these forums.

You can find an easy phrasing of the Riemann hypothesis in this section:
https://www.physicsforums.com/insig...ans-sum/#Zeros-and-Poles-of-the-zeta-Function
All you need to know is that $s=a+ib$ is a complex number and $\mathfrak{R}(s)=a$ denotes its real part, and $\Gamma(s)$ denotes the generalization of $\Gamma(n)=1\cdot 2\cdot 3\cdot \ldots\cdot (n-1)$ to complex numbers by setting
\begin{align*}\dfrac{n!n^s}{s\cdot (s+1)\cdot \ldots\cdot (s+n)} \stackrel{n\to \infty }{\longrightarrow } \Gamma(s)\end{align*}

The connection to prime numbers is basically due to the formula
$$
\zeta(s)=\left(\dfrac{1}{1-\dfrac{1}{2^s}}\right)\cdot \left(\dfrac{1}{1-\dfrac{1}{3^s}}\right)\cdot \left(\dfrac{1}{1-\dfrac{1}{5^s}}\right)\cdot \left(\dfrac{1}{1-\dfrac{1}{7^s}}\right)\cdot\left(\dfrac{1}{1-\dfrac{1}{11^s}}\right)\cdots
$$
which runs over all prime numbers. Hence, the distribution of the prime numbers has a direct impact on the value of the Riemannian Zeta-function. However, the complex power ${.}^s$ complicates things so that $\zeta(s)$ can take the value $0$ which it does not if $s$ is real. This is because the product runs over all primes and is therefore a limit. That means, if the product gets smaller and smaller, the limit will be zero, even if the single factors are not. That's the same as in the sequence $(1/2,1/3,1/4,1/5,1/6,\ldots)$. The single sequence members are positive, but their limit is zero since they get arbitrarily close to zero.

The Riemann hypothesis states that $\zeta(a+ib)$ has no zeros if $1>a>1/2.$ Now, the magic with the prime numbers is hidden in the equivalent formulation of this hypothesis that says: The Riemann hypothesis is true, if and only if the prime number counting function $\pi(x)=\{\text{number of primes smaller than } x\}$ is given by the formula $\pi(x)= f(x)+C\cdot \sqrt{x}+r(x),$ where $f(x)$ is a certain function, namely the so called integral logarithm, $C$ is some constant, and $r(x)$ an arbitrary small rest, an error term.

Of course, the work lies in the proof that those statements, the Riemann hypothesis and the distribution of primes, are equivalent. I have cited papers in which you can find this proof, but this requires mathematical techniques on a higher level, namely integration, and differentiation. See, e.g., https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf, which basically follows Riemann's original paper https://de.wikisource.org/wiki/Über_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Größe.

So your question can be rephrased as: Can you explain the eight pages of Riemann's paper about the distribution of prime numbers in simple words? The answer is yes, but it is not easy and certainly not short. Sciences are also always languages, i.e., a terminology that allows the description of complicated facts in short terms. Deassembling it into fundamental pieces is usually a mess, and the longer, the harder the problem is. And here, we have a hard (unsolved) problem whose formulation alone already takes higher mathematics. What you can take from that discussion is that Riemann's original work was about the prime number counting function, and the statement about the zeros of the zeta-function was a byproduct. This can be forgotten if we start at the other end with the Riemann hypothesis.

 

[martinbn]

You may have to accept that not evey topic in mathematics can be explained at a hight school level.

 

[weirdoguy]

in mathematics

And in physics likewise.

 

[fresh_42]

Here is an interesting quote by Albert Ingham that I have found on the German Wikipedia page about the Zeta-function:

One might consider the proof of the prime number theorem […] by de la Vallée, Poussin, and Hadamard unsatisfactory, because it introduces concepts that are very far removed from the original problem. Therefore, it is only natural to ask for a proof that does not depend on the theory of functions of a complex variable. To this, we must reply that no such proof is currently known. We can indeed go further and say that it is unlikely that a truly real proof will be found; at least, this is unlikely as long as the theory is based on Euler's identity. For every known proof of the prime number theorem relies on a certain property of the complex zeros of $\zeta(s),$ and conversely, this property is a simple consequence of the prime number theorem itself. It therefore seems clear that this property must be used explicitly or implicitly in any proof based on $\zeta(s),$ and one does not see how a proof can be carried out using only the real values of $s.$

Also interesting, this quote by Alte Selberg that I have found on the German Wikipedia page about the history of the Riemann hypothesis:

I think we're so keen to believe the Riemann Hypothesis is correct because it's the most beautiful and simplest distribution of zeros we can imagine. There's this symmetry along the critical line. Furthermore, it would result in the most natural distribution of prime numbers. Somehow, we'd like to believe that at least something in this universe should be correct.

I recommend reading that page:
https://de.wikipedia.org/wiki/Riemannsche_Vermutung#Geschichte
If you cannot read German, then Google Chrome translates it for you (right-click on the page). It is different from the English page on the Riemann hypothesis, which doesn't explicitly contain a historical perspective.

A personal note: I do not share the opinion that some things cannot be explained at a high school level. I understand that few actually can, but I don't think it is impossible. To me, it's just a matter of time, willingness, and effort. The question is not if, since any explanation could, for example, bridge the gap of knowledge by teaching it, rather it is a matter of how detailed it is requested to be.

 

[mathwonk]

Here are some naive comments I made over 20 years ago as AMS reviewer of apparently the first English translation of Riemann's "Werke": (I hope they are essentially correct).

"Riemann's philosophy that a meromorphic function is a global
object, associated with its maximal domain, and determined in any
subregion, "explains" why the analytic continuation of the zeta function
and the Riemann hypothesis help understand primes. I.e. Euler's product
formula shows the sequence of primes determines the zeta function, and
such functions are understood by their zeroes and poles, so the location
of zeroes must be intimately connected with the distribution of primes!

More precisely, in VII Riemann says Gauss's logarithmic integral
Li(x) actually approximates the number ( x ) of primes less than x, plus
1/2 the number of prime squares, plus 1/3 the number of prime cubes,
etc..., hence over - estimates π(x). He inverts this relation, obtaining a
series of terms Li(x^[1/n]) as a better approximation to (x), whose proof
apparently requires settling the famous "hypothesis"."

 

[phyzguy]

I think a key part of understanding the Riemann hypothesis is understanding why the Euler formula is true. Wikipedia has an excellent page with some nice graphics showing why this is true.

 

[fresh_42]

Here is the formal proof (first one)
\begin{align*}\left(1-\dfrac{1}{p^s}\right)\zeta(s)&=\sum_{n=1}^\infty \dfrac{1}{n^s}-\sum_{n=1}^\infty \dfrac{1}{(pn)^s} =\sum_{\stackrel{n=1}{n\not\in p\mathbb{Z}}}^\infty \dfrac{1}{n^s}=\sum_{\stackrel{n=1}{p\nmid n}}^\infty \dfrac{1}{n^s}\\\prod_{p\in\mathbb{P}}\left(1-\dfrac{1}{p^s}\right)\zeta(s)&=\sum_{\stackrel{n=1}{n\not\equiv 0\mod p\,\forall \,p\in \mathbb{P}}}^\infty \dfrac{1}{n^s}=\sum_{n\in \{1\}}\dfrac{1}{n^s}=1\\\zeta(s)&=1/\prod_{p\in\mathbb{P}}\left(1-\dfrac{1}{p^s}\right)=\prod_{p\in\mathbb{P}}\left(1/\left(1-\dfrac{1}{p^s}\right)\right)=\prod_{p\in\mathbb{P}}\dfrac{1}{1-p^{-s}}\end{align*}

Source: https://www.physicsforums.com/insig...nn-hypothesis-and-ramanujans-sum/#toggle-id-2

It is not really difficult.

 

[martinbn]

...I.e. Euler's product
formula shows the sequence of primes determines the zeta function, and
such functions are understood by their zeroes and poles, so the location
of zeroes must be intimately connected with the distribution of primes!

So is the pole. It implies that there are infinitely many primes.

 

[TensorCalculus]

3Blue1Brown has an excellent video on it that gives a high level overview without needing much advanced mathematics (a 12 year old was able to understand it, but it explained it nonetheless, even if the explanation was a quite simplified). I can't provide the link right now due to Internet problems (I can give to you tomorrow) but if you search for it you should find it on YouTube. Alternatively, there's a very very basic explanation understandable by anyone provided by Matt Parker in his book things to make and do in the fourth dimension but I highly doubt that it will fully satisfy your curiosity - it does not go into detail at all but instead aims to present it in a way where people can just barely get to grips with what is going on.

 

[pbuk]

3Blue1Brown has an excellent video on it

Link: https://www.3blue1brown.com/lessons/zeta

 

[TensorCalculus]

Link: https://www.3blue1brown.com/lessons/zeta

Thanks. I forgot to do that, whoops.