Last year, I found very helpful lecture notes on analytic number theory and the Riemann Zeta function on Otto Forster‘s website. You might actually know this author by his (at least in Germany) very widely used real analysis text book series.
He has on his website several nicely typeset sets of lecture notes (but be warned that many of them are in German). Those lecture notes that I would like to point out here are:
These lectures notes overlap in some parts. They helped me a lot in understanding the rough outline of the usually taught proof of the prime number theorem and the concepts in analytic number theory in general. They also gave me a first glimpse of how the Riemann Zeta function can be approached and demystified.
Unfortunately, the chapter on Euler-Maclaurin summation is not available online. That’s why I recently bought H. M. Edward’s book “Riemann’s Zeta Function” about which I will write more in future posts.
I have compiled a YouTube playlist that contains short video portraits of famous mathematicians. The videos in this playlist have always motivated me and for some reason I like watching videos like this a lot. This sometimes feels like it’s a strange obsession of mine, reminds me a bit of a teenager who obsesses with his favourite internet personality (or TV personality when i was still a teenager). But on the other hand I realized that these videos also contain a few memorable quotes that I would like to point out here.
Of the 2018 portraits of the Fields Medal recipients, the first one in the playlist is of Peter Scholze. This video contains the quote
“There’s always an infinite number of problems to be solved in mathematics. […] whenever you solve one there are ten more coming.”
That quote certainly resonates with me because I have also made the experience that solving/understanding one problem just naturally leads one to asking more questions.
In Caucher Birkar’s video, I very much liked the quote
“Reading mathematics […] is like going to a […] beautiful town. When you walk around you see monuments, you see beautiful architecture. And that’s like the first stage, where you just see what other people have created. The second stage is like, if I suddenly have wings and I fly over a city and I can see a lot more than before.”
In Akshay Venkatesh’s video, my favourite quote is this:
“When I was maybe around seven, I remember I had this spiral notebook and I had just learned about binary. And I remember writing, in red, various numbers translated into binary. I think just manipulating numbers makes me feel happy.”
This quote shows in a somewhat sincere way that in order to make some progress (in mathematics or in any field really) one needs to enjoy the process, the working with the tools (or the people involved). In any case, it must be fun to do it. This idea can really be translated to every person and every type of human activity. It’s also something that Martin Hairer mentions in his 2014 Fields Medal portrait by saying that one must be “genuinely interested”.
While Manjul Bhargava mentions the importance of playing with toys to somehow stimulate the mind, Jacob Lurie, in his MacArthur Fellow portrait, likens mathematics to “a giant playground, filled with all kinds of toys that the human mind can play with. But many of these toys have very long operating manuals”. That’s certainly a cool metaphor. And in the comments to this video some person said this: “Many of these toys have very long operating manuals. Yeah, ‘Higher Topos Theory’ is like 900+ pages long…” (Higher Topos Theory being a book/article of Lurie’s). But the point that Lurie is trying to make is that some mathematical phenomena are not that difficult to explain and presenting them in engaging ways might certainly help lightening up common mathematics classes (below university level).
By the way, if you like the videos in my playlist, you might also be interested in the movies that George Csicsery made produced by ZALA films. Among other documentaries, he made films about Yitang Zhang, Paul Erdős and Maryam Mirzakhani that are approximately an hour long.
And last but not least I must say that I like watching all scenes in all of these videos that contain footage of notebooks. I find it fascinating to see what and how other people make notes and mostly it’s just beautiful to look at them.
In this post, I would like to demonstrate how to calculate the Bernoulli numbers. The Bernoulli numbers are a sequence of rational numbers and they’re usually denoted . If you want a list of their values, you can go to the OEIS … oh, wait, you can’t actually look them up directly in the OEIS because they are not integers. But, hey, since they are rational numbers, you can just look up their numerators and denominators separately 😆.
The reason why I am interested in them is that they can be used to evaluate the Riemann Zeta function, as is explained in this paper. This great mathologer video gives another reason why one might be interested in the Bernoulli numbers and also explains how Bernoulli found them in the first place. This will help you understand this post’s featured image. Eventually, the mathologer video leads up to the Euler-Maclaurin formula, which is also the underlying reason why the Bernoulli numbers are useful for evaluating the Riemann Zeta function.
In the following we will aside the genesis and applications of the Bernoulli numbers and we will not discuss the convergence of the series through which we define them. We will assume that this has been taken care of. This post is about the numerical calculation of the Bernoulli numbers.
1. The arithmetic
The Bernoulli numbers are defined via the power series of the following fraction:
To get a handle on the Bernoulli numbers, which are the parts of the coefficients denoted by , it helps to observe the following expansion of the inverse fraction first:
Here we have only used the well-known series expansion of the exponential function and then multiplied out the factor and re-written the result as a sum.
No, if we multiply both fractions, we get the following relation:
and we just imagine that this product has its own power series expansion . We know that this series has to be equal to 1 regardless of the value of . And it is not hard to find out that they obey the following relation:
To obtain this expression of the we have used what is commonly known as the Cauchy product formula. But this just means that if we have to sums and and their prodcut expansion then we just “collect” all terms from the original two series where . For example:
Here, we have two possible combinations of a-b-factors in front of the linear term(s). Therefore, the coefficient in the prodcut’s expansion needs to sum over those. And in the infinite case we can ignore the fact that our toy example has a larger highes term in .
Ok, back to our main result:
Here, we can easily check (if we remember that ) that . Since the total product is identically equal to 1 and is already equal to 1, we know that .
2. The triangle
Ok, so far we have managed to get the first Bernoulli number . But given the fact that all must be zero, we can write down the equations for the next few Bernoulli numbers. Let’s do this and then discuss how to solve that:
Now, if we stare at this for a while, we might actually realize that the coefficients of the in this list of equations are actually the entries in Pascal’s triangle. That is, we are laying out all binomial coefficients where the “numerator” of the binomial coefficient increases row-wise and the “denominator” increases within a row.
Therefore, the coefficients of the in any given row are obtained by summing the “neighbouring” coefficients of the previous row, just as one would do it in Pascal’s triangle. The first few Bernoulli numbers are thus:
Thre first few Bernoulli numbers
Note that the Bernoulli numbers with odd index greater than 1 are all zero, not just those first two of this kind that can be seen in this table. Wikipedia has a section on why this is the case.
3. The code
For my personal usage I have compiled a small Python function that returns the k-th Bernoulli number. I have not optimized this in any way, for example, I have not added an early exit condition for odd k. I have also not added any caching here, so all numbers will always be calculated from the tip of the triangle.
def my_bernoulli(k): coefficients =  B =  for i in range(1, k+1): new_coefficients = coefficients coefficients = coefficients +  new_coefficients = new_coefficients +  for j in range(1, i+1): new_coefficients[j] = coefficients[j-1] + coefficients[j] sum = -B for j in range(1, i): sum -= new_coefficients[j] * B[j] B.append(sum/new_coefficients[i]) coefficients = new_coefficients
The channel is created by mathematician postdoc Maria Yakerson and she posts mostly interviews with fellow mathematicians. Occasionally, she discusses her own feelings around her career as a professional mathematician. In one video, she mentioned that one reason for her to start this channel is that she feels like an oustider in her discipline from time to time. The other reason being of course the joy she finds in learning from her interviewees. What I like about the channel is that the interview partners are truly divers.
The interviews usually contain some questions about how the interviewee got into mathematics, about their struggles in mathematics, about fears and also about some advice to fellow mathematicians. In the case of older/established mathematicians she might also ask them for advice to younger colleagues. You might also find a decent amount of anecdotes, partially related to famous names, in there.
It also occurred to me that many of the interviewees seem to be quite reserved. But I imagine that I would have a hard time myself to open up if I don’t know the interviewer very well and if my interview will be published on YouTube. On the other hand Maria does make the impression or convey the feeling that she will not judge your answers too much which creates an emotionally safe space which is an important ingredient for such a project.
I guess one clear commonality among almost all interviewees is that they all struggle in their job (which probably most of us do sometimes) and that they all feel “stupid” sometimes. But I can hardly imagine that their feeling stupid is due to a selection bias on part of the interviewer, it is probably just part of the game. One could even go so far as to say that if one person, let’s call her Alice, feels stupid in some talk or conversation with another person, say Bob, then it’s either because of a lack of empathy on Bob’s part of because Alice just chose the wrong talk to sit in on. Even the latter could have been only caused by a lack of information about the talk’s content or due to social pressure or whatever. So in almost no case was any actual mathematical stupidity involved.