Commentary for "What is the Riemann Hypothesis?"

Staff member

Jameson

Staff member
When I was 15 or so I became very interested in the Riemann Hypothesis and the other Millennium Problems from the Clay Institute, so I bought a book called Prime Obsession that covered the RH from a historical point of view instead of a mathematical one. It is a good book for non-mathematicians that reasonably introduces concepts you can't expect an average person to know. However, this book might mislead readers into thinking that the RH is much easier to follow than it truly is since the actual proposition can be stated very succinctly. It reminds me of when I hear someone discussing String Theory with way too much confidence after watching a 30 minute TV show on The History Channel.

The summary Balarka is writing seems to be at a good level for those working at the undergrad level but isn't overwhelming to read either. If I don't follow some part I am at least aware of the topic that needs more study. I've skimmed the whole thread but will be reading in detail very soon. Well done, Balarka .

mathbalarka

Well-known member
MHB Math Helper
Hey, Jameson, thanks very much - I am glad to hear that it was of some use. My intention at first was to use no advanced stuff like complex analysis however I soon reckoned that I couldn't ignore it. Complex analysis and number theory are intimately connected, much much more connected than real analysis is. It'd be an oversimplification of the problem if I didn't use CA.

However, the only _assumed_ result I have used is that Perron's formula. If anyone goes through the post rigorously and finds curiosity about Perron, I can prove it for them. It requires a bit contour manipulation and some hard-core estimation. In it's most general and powerful form, it is

$$\sum_{n \leq x} \frac{a_n}{n^s} = \frac1{2\pi i} \int_{c - iT}^{c + iT} \sum_{n=1}^\infty \frac{a_n}{n^{s+\omega}} \frac{x^\omega}{\omega} \mathrm{d}\omega + O \left ( \frac{x^c}{T(\sigma + c - 1)^\alpha} \right) + O \left ( \frac{f(2x) x^{1-\sigma} \log x}{T} \right ) + O \left ( \frac{f(N) x^{1-\sigma}}{T|x - N|} \right )$$

Where $a_n = O(f(n))$, $\psi(n)$ not decreasing, $\sum_{n \leq x} |a_n|/n^\sigma = O((\sigma - 1)^{-\alpha})$ in the neighborhood of $\sigma = 1$, some $c > 0$ such that $\sigma + c > 1$ and a noninteger $x$ with $N = \text{int}(x)$.

Letting $T \to \infty$ and $s = 1$, however, one arrives at the weaker formula shown in wiki, weaker in the sense that the absolute convergence may not be verified anymore, neither of the sum nor of the integral (the zeta zero sum in the explicit formula for $\psi(x)$ a couple posts back is, for example, not absolutely converging and it'd be a capital mistake to compute the sum without ordering the roots, see Riemann's trap )

I've also explained some elementary version of RH, like the moebius-RH : $M(x) = O(x^{1/2+\epsilon})$ and it's probabilistic interpretation, the $\psi$-RH : $|\psi - x| \leq k \cdot \sqrt{x}^{1/2} \log^2 x$ for some $k > 0$ which is in fact the tightest bound, following Littlewood's result. I feel a relation to prime gaps (can be probabilistically explained!) would have been interesting but unfortunately I haven't studied it at all. There are a few plots and a table (and _will be_ computational results) too, so some intuition is present. In the whole, I believe the intuitive and the unintuitive part cancels each other out

That reminds me, I think I have to add a Venn diagram to the analytic continuation post : a lot of hubbub about analytic continuation is going round. Some are getting the wrong idea, and some never looking back at it. While analytic continuation is a delicate business in complex analysis (indeed, they are the starting point of Riemannsurfaces) I have really used nothing than basics of calculus in that post.

Finally, I'd like to have a few people reading it thoroughly and asking me anytime about anything that isn't explained well in the thread.

mathbalarka

Well-known member
MHB Math Helper
Announcement : I have just finished the note, there will be no further posts.

The solutions are expected to be posted a few weeks later, it maybe further delayed on request from members (due to reasons like trying a problem in there, etc). Solution/ideas about any of the problems are most certainly welcome.

Pointing out any kind of typo (there are a lot of them!) are welcome, especially $\LaTeX$ ones.

Any kind of questions regarding the content will definitely be answered by the author (that's me).