But what is the Riemann zeta function? Visualizing analytic continuation
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it whenever someone asks about analytic continuation, the -1/12 sum, or what the zeta function looks like in the complex plane.
Recommended by 13 people across 13 different months, 2014 to 2022.
Why Hacker News recommends it
HN praises its visualizations, its handling of the -1/12 result, and its ending, and often prefers it to other explanations of the topic.
Recommended by 6 people across 5 different months, 2017 to 2023.
What it is
3Blue1Brown visualizes the Riemann zeta function and the analytic continuation that extends it beyond where its series converges, building toward the Riemann hypothesis.
Cited in 1 Wikipedia article, including Riemann zeta function.
What people said
Strong rec JoseOrtiz3 · 2018-02 · Mathematics Stack Exchange · on “why is that $1+2+3+4+5+6+…. = -1/12$?”
Short, wonderful coverage of this topic by 3Blue1Brown (also tons of other cool math things on his channel): [this video]
Strong rec laurilove · 2018-05 · Hacker News · on “An Interesting Pattern in the Prime Numbers: Parallax Compression”
Appropriate juncture to repost this: [this video] "Visualizing the Riemann zeta function and analytic continuation" (Watch the last 30s or so if nothing else. You'll not regret it.) There is a kind of 'real' Ural spiral [or really a kind of chaotically decaying 'orbit'] related dynamic that gives the prime numbers (assuming the RH.) Conceivable that OP's pattern might have some relation to this dynamic, and fun pondering to boot.
Strong rec st1x7 · 2021-01 · Hacker News · on “How I learned to love and fear the Riemann Hypothesis”
I don't think it comes close to 3blue1brown's video on the same topic - [this video] Still nice of the video in the original post to give a bit more of a historical context, instead of focusing entirely on the mathematics. It could have done without the building analogy around the beginning though, that didn't come up later and was just distracting.
Strong rec Eli Yablon · 2022-05 · Mathematics Stack Exchange · on “Riemann Hypothesis and Electric Dipole”
I recently watched 3Blue1Brown's video [this video] on the Riemann zeta function (excellent video by the way), and basically he shows what analytic continuation is and how it relates to the function. In order to visualize the Riemann zeta function being applied to the complex plane, he takes the horizontal and vertical lines on the regular complex plane and then draws what they look like after the Riemann zeta function has been applied. [LINK] I happened to notice that this image looks like the equipotential lines and field lines in an electric dipole [LINK] . This leads me to two questions: Is the picture from 3Blue1Brown's video the same thing as equipotential lines and field lines in an electr
Recommends Nick · 2014-01 · Mathematics Stack Exchange · on “To sum $1+2+3+\cdots$ to $-\frac1{12}$”
[19/3/19] Update - ref. 3b1b - Visualizing the Riemann hypothesis and analytic continuation : [this video] - this piece demonstrates utmost clarity on $\zeta(-1)$ evaluation. :: An explanation of how this is true and why and where it is useful and significant is summarized in 24 by John Baez (as part of the 19/09/08 Rankin Lectures) [LINK] and yes, indeed this is weird [LINK] . The following is the same Euler's method described by Luboš Motl and Matt E in the other answers. Let us consider the infinite geometric progression " $1, x, x^2, x^3, x^4, \dots$ " The sum of this progression can be found by the formula $S_\infty = \large \frac{a}{1-r}$ , $$\implies 1 + x + x^2 + x^3 + x^4 + \dots = \frac
Recommends LoMaPh · 2014-04 · Mathematics Stack Exchange · on “Why is the sum over all positive integers equal to -1/12?”
Usually we do not assign a value to divergent series. The assertion that these kinds of series are divergent means that these kinds of series do not have sums. Now if we want to assign a sum (in some sense) to divergent series, what these sums "should" be and how they should be calculated? Obviously we are going out of customary mathematics. We are extending it to "illegal" cases, and it is not a surprise that we get odd results. G.H. Hardy's "Divergent Series" is a good reference. You can find more discussion on this subject there. I also found this [this video] video very helpful.
Recommends Alex Jones · 2016-12 · Mathematics Stack Exchange · on “Relation between $\zeta(s), \ Re(s) < 1$ and the summation $\sum_{k=1}^\infty k^{-s}$”
This may or may not answer your question, but I think a lot of insight and intuition about the nature of this function and analytic continuation in general can be found here: [this video]
Recommends vq · 2017-01 · Hacker News · on “The Riemann Hypothesis, explained”
3Blue1Brown recently did a video on the Riemann zeta function: [this video] It's not as in depth but it has some helpful visualisations that I've never seen elsewhere.
Recommends Agno · 2017-01 · Mathematics Stack Exchange · on “Motivations and applications for the functional equation of $\zeta$”
This is not an answer, but I believe that one of the obvious applications of the functional equation is that it provides a full analytic continuation of $\sum_{n=1}^{\infty}\frac{1}{n^s}$ into the domain $\Re(s) <1$. The 'appealing' side of this continuation is beautifully visualised in this video (start e.g. at 10:00): [this video]
Recommends Stefan Gruenwald · 2017-03 · Mathematics Stack Exchange · on “Checking understanding of summations of divergent series”
You get this result by analytic continuation of the Riemann zeta function. Watch this video to see it graphically: [this video]