I’m giving a talk today on my recent and forthcoming work in collaboration with Theresa Anderson, Ayla Gafni, Robert Lemke Oliver, George Shakan, Frank Thorne, Jiuya Wang, and Ruixiang Zhang. The slides for my talk can be found here.

This talk includes some discussion of our paper to appear in IMRN (link to the arXiv version, which is mostly the same as what will be published). (See also my previous discussion on this paper). But I’ll note that in this talk I lean towards a few ideas that did not make it into the paper, but which we are using in current work.

In particular, in our paper we don’t need to use group actions or classify orbit sizes, but it turns out that this is a very strong idea! I’ll note that in a very particular case, Thorne and Taniguchi have applied this type of orbit counting method in their paper “Orbital exponential sums for prehomogeneous vector spaces” to gain extremely strong, specific understanding of Fourier transform for their application.

This summer, I proposed a research project for PROMYS (the PROgram in Mathematics for Young Scientists), a six-week intensive summer program at Boston University where highly motivated high school students explore mathematics. Three students (Nir Elber, Raymond Feng, and Henry Xie) chose to work on this project, and previous PROMYSer Anupam Datta gave additional guidance. Their summary of their findings can be found here. (UPDATE: a version of this now appears on the arXiv too).

Here I briefly describe the project and the work of Nir, Raymond, and Henry.

The project was organized around understanding why the following picture has so much structure.

Fundamentally, this image depicts differences between sums related to primes. Let $p_n$ denote the $n$th prime. It follows from the Prime Number Theorem that $p_n \approx n \log n$, and thus that $n p_n \approx n^2 \log n$. One can also show that $$ \sum_{m \leq n} p_m \approx \frac{1}{2} n^2 \log n,$$ and thus we should have that $$ \frac{n p_n}{\sum_{m \leq n} p_m} \to 2.\tag{1}$$

The vertical axis in the image above examines differences between consecutive $n$ in $(1)$ (in log scale), while the horizontal axis gives $n$ (also in log scale).

The fact that $(1) \to 0$ corresponds to the overall downwards trend in the graph. But there is so much more structure! Why do the points fall into “troughs” or along “curtains”? Does each line mean something?

In this version, I’ve colored differences coming from when $p_n$ is a twin prime (in blue), a cousin prime (in green), a sexy prime (in red), or a prime $p$ such that the next prime is $p+8$ (in cyan). The first dot is black because it comes from $2$. The next two correspond to $3$ and $5$ (both twin primes), and the fourth dot corresponds to $7$ and is green because the next prime after $7$ is $11$, and so on.

This is a strong hint at distributional aspects alluded to within the plots.

Nir, Raymond, and Henry proved many things! They quantified the rate of convergence in $(1)$ and thus quantified the guaranteed downward trend in the images and found images that better convey the structure of what’s going on better. I was already very impressed, but then they branched out and studied more!

Cramér’s Model

We chose to investigate a nuanced question: what aspects of the initial plots depend strongly on the fact that the underlying data consists of primes, and what aspects depend only on the fact that the underlying data consists of integers with the same density as the primes?

To study this, one can create a new set of distinguished elements called Promys Primes (PPrimes) with the same density as true primes using probabilistic ideas of Cramér. Let’s call $2$ and $3$ PPrimes, and then for each odd $m \geq 5$, we call $m$ a PPrime with probability $2 / \log m$. Do this for a large sequence of $m$, and we get a collection of PPrimes that has (with very high probability) the same density as true primes, but none of the multiplicative structure.

It turns out that for sets of PPrimes, there are analogous pictures and the asymptotics are even better! This is in section 3 of their write-up.

Gaussian Integers

We also thought to study analogous situations in related sets of primes, such as the Gaussian integers. Recall that the Gaussian integers $\mathbb{Z}[i] = \{ a + bi : a, b \in \mathbb{Z} \}$ are a unique factorization domain and have a rich theory of primes. Sometimes this theory is very similar to the standard theory of primes over $\mathbb{Z}$. But there are challenges.

One significant challenge is that $\mathbb{C}$ is not ordered. A related challenge is that there are more units. Over $\mathbb{Z}$, both $2$ and $-2$ are primes, but we typically recognize $2$ as being more “simple”. For Gaussian primes, there isn’t such a choice; for example each of $1 + i, 1 – i, -1 + i, -1 – i$ are Gaussian primes, but none are more simple or fundamental than the others.

More concretely, one has to be careful even with how to define the “sum of the first $n$ primes”. One natural thought might be to sum all Gaussian primes $\pi$ that have norm up to $X$. But one can quickly see that this sum is $0$ for analogous reasons to why the sum of all the typical primes with absolute values up to $X$ must vanish ($p + -p = 0$). In the Gaussian case, it is also true that $$ \sum_{N(\pi) \leq X} \pi^2 = 0.$$

But they considered higher powers, where there aren’t trivial or obvious reasons for massive cancellation, and they showed that there is always nontrivial cancellation. This is interesting on its own!

Then they also constructed a mixture, a Cramér-type model for Gaussian primes and showed that one should expect nontrivial cancellation there for purely distributional reasons.

I leave the details to their write-up. But they’ve done great work, and I look forward to seeing what they come up with in the future.

I gave a talk on visualizations of modular forms made with Adam Sakareassen at Bridges 2021. This talk goes with our short article. In this talk, I describe the line of ideas going towards producing three dimensional visualizations of modular forms, which I like to call modular terrains. When we first wrote that talk, we were working towards the following video visualization.

We are now working in a few different directions, involving informational visualizations of different forms and different types of forms, as well as purely artistic visualizations.

I’ve recently been very fond of including renderings based on a picture of my wife and I in Iceland (from the beforetimes). This is us as a wallpaper (preserving many of the symmetries) for a particular modular form.

Yesterday I gave a talk at the University of Oregon Number Theory seminar on Visualizing Modular Forms. This is a spiritual successor to my paper on Visualizing modular forms that is to appear in Simons Symposia volume Arithmetic Geometry, Number Theory, and Computation.

I’ve worked with modular forms for almost 10 years now, but I’ve only known what a modular form looks like for about 2 years. In this talk, I explored visual representations of modular forms, with lots of examples.

I’m currently at an AIM workshop on Arithmetic Statistics, Discrete Restriction, and Fourier Analysis. This morning (AIM time)/afternoon (USEast time), I’ll be giving a talk on Lattice points and sums of Fourier Coefficients of modular forms.

The theme of this talk is embodied in the statement that several lattice counting problems like the Gauss circle problem are essentially the same as very modular-form-heavy problems, sometimes very closely similar and sometimes appearing slightly different.

In this talk, I describe several recent adventures, successes and travails, in my studies of problems related to the Gauss circle problem and the task of producing better bounds for the sum of the first several coefficients of holomorphic cuspforms.

I’ll note that various parts of this talk have appeared in several previous talks of mine, but since it’s the pandemic era this is the first time much of this has appeared in slides.

In a remarkable coincidence, I’m giving two talks on Maass forms today (after not giving any talks for 3 months). One of these was a chalk talk (or rather camera on pen on paper talk). My other talk can be found at https://davidlowryduda.com/static/Talks/ComputingMaass20/.

In this talk, I briefly describe how one goes about computing Maass forms for congruence subgroups of $\mathrm{SL}(2)$. This is a short and pointed exposition of ideas mostly found in papers of Hejhal and Fredrik Strömberg’s PhD thesis. More precise references are included at the end of the talk.

This amounts to a description of the idea of Hejhal’s algorithm on a congruence subgroup.

Side notes on revealjs

I decided to experiment a bit with this talk. This is not a TeX-Beamer talk (as is most common for math) — instead it’s a revealjs talk. I haven’t written a revealjs talk before, but it was surprisingly easy.

It took me more time than writing a beamer talk, most likely because I don’t have a good workflow with reveal and there were several times when I wanted to use nontrivial javascript capabilities. In particular, I wanted to have a few elements transition from one slide to the next (using the automatic transition capabilities).

At first, I had thought I would write in an intermediate markup format and then translate this into revealjs, but I quickly decided against that plan. The composition stage was a bit more annoying.

But I think the result is more appealing than a beamer talk, and it’s sufficiently interesting that I’ll revisit it later.

The current issue of the Proceedings of the Royal Society A^{1} features cover artwork made by Vikas Krishnamurthy, Miles Wheeler, Darren Crowdy, Adrian Constantin, and me.

A version of the cover pre-addition is the following.

This is based on the work in A transformation between stationary point vortex equilibria, which concerns solutions to Euler’s equation for inviscid (2D) fluid motion $$ \frac{\partial \mathbf{V}}{\partial t} + (\mathbf{V} \cdot \nabla) \mathbf{V} = – \frac{\nabla p}{p_0}, $$ where $\nabla = (\partial/\partial x, \partial / \partial y)$ is the 2D gradient operator. There is a notion of vortices for these systems, and the paper examines configurations of point vortices under certain idealized conditions that leads to particularly nice analysis. In the situation studied, one can sometimes begin with one configurations of point vortices and perform a transformation that yields another, bigger and more complicated configuration.

This is the situation depicted on the cover — begin with a simple configuration and iterate the process. The spiral shape was added afterwards and doesn’t describe underlying mathematical phenomena. The different colors of each vortex shows whether that vortex is a sink or a source, essentially.

I was told most of this after the fact by Miles — who researches fluid dynamics, is a friend from grad school, and was my coauthor on a paper about the mean value theorem. I do not typically think about fluid dynamics (and did not write the paper), and it’s a bit funny how I got involved in the production of this cover. But it was fun, and we produced many arresting images. In the future Miles and I intend to revisit these images and better describe how the various aspects of the image describes and reflects the underlying mathematical behavior.

As a fun aside — we didn’t only produce one image. We made many, and we made many configurations.^{2} In my work on visualizing modular forms, I developed a few techniques for color selection from matplotlib style colormaps, and produced several variants. I’ve collected a few of these below.

I recently gave a talk “at Dartmouth”^{1}. The focus of the talk was the (odd-indexed) Fibonacci zeta function:
$$ \sum_{n \geq 1} \frac{1}{F(2n-1)^s},$$
where $F(n)$ is the nth Fibonacci number. The theme is that the Fibonacci zeta function can be recognized as coming from an inner product of automorphic forms, and the continuation of the zeta function can be understood in terms of the spectral expansion of the associated automorphic forms.

This is a talk from ongoing research. I do not yet understand “what’s really going on”. But within the talk I describe a few different generalizations; firstly, there is a generalization to other zeta functions that can be viewed as traces of units on quadratic number fields, and secondly there is a generalization to quadratic forms recognizing solutions to Pell’s equation.

I intend to describe additional ideas from this talk in the coming months, as I figure out how pieces fit together. But for now, here are the slides.

In my previous note, we considered equidistribution of rational points on the circle $X^2 + Y^2 = 2$. This is but one of a large family of equidistribution results that I’m not particularly familiar with.

This note is the first in a series of notes dedicated to exploring this type of equidistribution visually. In this note, we will investigate a simpler case — rational points on the line.

When you order rational points on the circle $X^2 + Y^2 = 2$ by height, these points equidistribute.

Stated differently, suppose that $I$ is an arc on the circle $X^2 + Y^2 = 2$. Then asymptotically, the number of rational points on the arc $I$ with height bounded by a number $H$ is equal to what you would expect if $\lvert I\rvert /2\sqrt{2}\pi$ of all points with height up to $H$ were on this arc. Here, $\lvert I\rvert /2\sqrt{2}\pi$ the ratio of the arclength of the arc $I$ with the total circumference of the circle.

This only makes sense if we define the height of a rational point on the circle. Given a point $(a/c, b/c)$ (written in least terms) on the circle, we define the height of this point to be $c$.

In forthcoming work with my frequent collaborators Chan Ieong Kuan, Thomas Hulse, and Alexander Walker, we count three term arithmetic progressions of squares. If $C^2 – B^2 = B^2 – A^2$, then clearly $A^2 + C^2 = 2B^2$, and thus a 3AP of squares corresponds to a rational point on the circle $X^2 + Y^2 = 2$. We compare one of our results to what you would expect from equidistribution. From general principles, we expected such equidistribution to be true. But I wasn’t sure how to prove it.

With helpful assistance from Noam Elkies, Emmanuel Peyre, and John Voight (who each immediately knew how to prove this), I learned how to prove this fact.

David Lowry-Duda October 19, 2021 at 7:14 pm on An Intuitive Overview of Taylor SeriesHi Bob! The behavior of $c$ is actually quite subtle. It's not true that $c$ is actually a constant. For "nice" functions, what is true is that the mean value $c$ varies continuously over intervals (except at finitely many points). Combined with a form of Darboux's theorem (stating that every function that is the derivative of another function has the ---

Bob October 18, 2021 at 6:09 pm on An Intuitive Overview of Taylor SeriesIt's been a while since there was a comment here, but I'll give it a try in the hope for a response. Your proof gives me a sense of validation in my own work, although I still have questions. I wrote a very similar idea here: https://math.stackexchange.com/q/4277898/225136 . As you can see from a comment that someone left, the proof ---

Yavor Kirov April 14, 2021 at 3:24 am on Long live opalstack!Thabk you! I was looking for an opinion of a long-term user of Opalstack so your answer was exactly what I needed.

David Lowry-Duda April 13, 2021 at 9:32 pm on Long live opalstack!Hi Yavor, I'm very happy with Opalstack. After the initial transition, there have been almost no problems. I'll note that for a few months, opalstack's email setup was rocky, but they figured that out. I'd recommend it. They've been pretty responsive to problems, too. You can see what's troubling people now on their community/support forum: https://community.opalstack.com/. I know a somewhat ---

Yavor Kirov April 13, 2021 at 6:12 pm on Long live opalstack!Hello David, Would you please be so kind to tell us (me) of your (further) experience with Opalstack so far? I myself had to migrate away from WebFaction and I am wondering if now migrating some of my sites to Opalstack is a good idea. Best regards :)

David Lowry-Duda February 25, 2021 at 6:24 pm on phase_mag_plot: a sage package for plotting complex functionsI'm glad to hear it! I would also read issues or pull requests on github. I think it would not be particularly hard to incorporate this into official sage. I have a local build of sage that includes a variant of phase_mag_plot as the default complex_plot. There are a few additional usability things I need to do before submitting this ---

Jan van Delden February 25, 2021 at 5:07 pm on phase_mag_plot: a sage package for plotting complex functionsI took the liberty to download your SageMath module and changed a few options. Since I was not completely clear on the interaction between lightness and color (defined by the argument of the function to be displayed) I decided to weigh these differently. Color, from the colorwheel by 0.8 and lightness by 0.2. I computed the color first and weighed ---

eliot December 23, 2020 at 4:07 pm on Trigonometric and related substitutions in integralsFor Euler substitutions, do you mind explaining the reasoning for the t term in x*sqrt(a) + t. Is there anywhere else I can read up on this? Thanks for this post!

David Lowry-Duda August 7, 2020 at 4:15 pm on AboutThat's great! If there are particular plots that you're interested in, let me know. But I (very recently) posted a library for making the plots https://davidlowryduda.com/phase_mag_plot-a-sage-package-for-plotting-complex-functions/ Alternately, I make sage/python notebooks available for many of my visualizations. These are usually linked to within the post. If you see something that you want to replicate and find the code, leave a ---