Category Archives: Mathematics

Trace form 3.32.a.a

When asked if I might contribute an image for MSRI program 332, I thought it would be fun to investigate a modular form with a label roughly formed from the program number, 332. We investigate the trace form 3.32.a.a.

The space of weight $32$ modular forms on $\Gamma_0(3)$ with trivial central character is an $11$-dimensional vector space. The subspace of newforms is a $5$-dimensional vector space.

These newforms break down into two groups: the two embeddings of an abstract newform whose coefficients lie in a quadratic field, and the three embeddings of an abstract newform whose coefficients lie in a cubic field. The label 3.32.a.a is a label for the two newforms with coefficients in a quadratic field.

These images are for the trace form, made by summing the two conjugate newforms in 3.32.a.a. This trace form is a newform of weight $32$ on $\Gamma_1(3)$.

Each modular form is naturally defined on the upper half-plane. In these images, the upper half-plane has been mapped to the unit disk. This mapping is uniquely specified by the following pieces of information: the real line $y = 0$ in the plane is mapped to the boundary of the disk, and the three points $(0, i, \infty)$ map to the (bottom, center, top) of the disk.

This is a relatively high weight modular form, meaning that magnitudes can change very quickly. In the contoured image, each contour indicates a multiplicative change in elevation: points on one contour are $32$ times larger or smaller than points on adjacent contours.

I have a bit more about this and related visualizations on my visualization site.

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Slides from a talk on Half Integral Weight Dirichlet Series

On Thursday, 18 March, I gave a talk on half-integral weight Dirichlet series at the Ole Miss number theory seminar.

This talk is a description of ongoing explicit computational experimentation with Mehmet Kiral, Tom Hulse, and Li-Mei Lim on various aspects of half-integral weight modular forms and their Dirichlet series.

These Dirichlet series behave like typical beautiful automorphic L-functions in many ways, but are very different in other ways.

The first third of the talk is largely about the “typical” story. The general definitions are abstractions designed around the objects that number theorists have been playing with, and we also briefly touch on some of these examples to have an image in mind.

The second third is mostly about how half-integral weight Dirichlet series aren’t quite as well-behaved as L-functions associated to GL(2) automorphic forms, but sufficiently well-behaved to be comprehendable. Unlike the case of a full-integral weight modular form, there isn’t a canonical choice of “nice” forms to study, but we identify a particular set of forms with symmetric functional equations to study. There are several small details that can be considered here, and I largely ignore them for this talk. This is something that I hope to return to in the future.

In the final third of the talk, we examine the behavior and zeros of a handful of half-integral weight Dirichlet series. There are plots of zeros, including a plot of approximately the first 150k zeros of one particular form. These are also interesting, and I intend to investigate and describe these more on this site later.

The slides for this talk are available here.

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A balancing act in “Uniform bounds for lattice point counting”

I was recently examining a technical hurdle in my project on “Uniform bounds for lattice point counting and partial sums of zeta functions” with Takashi Taniguchi and Frank Thorne. There is a version on the arxiv, but it currently has a mistake in its handling of bounds for small $X$.

In this note, I describe an aspect of this paper that I found surprising. In fact, I’ve found it continually surprising, as I’ve reproven it to myself three times now, I think. By writing this here and in my note system, I hope to perhaps remember this better.

Landau’s Method

In this paper, we revisit an application of “Landau’s Method” to estimate partial sums of coefficients of Dirichlet series. We model this paper off of an earlier application by Chandrasakharan and Narasimhan, except that we explicitly track dependence of the several implicit constants and we prove these results uniformly for all partial sums, as opposed to sufficiently large partial sums.

The only structure is that we have a Dirichlet series $\phi(s)$, some Gamma factors $\Delta(s)$, and a functional equation of the shape $$ \phi(s) \Delta(s) = \psi(s) \Delta(1-s). $$ This is relatively structureless, and correspondingly our attack is very general. We use some smoothed approximation to the sum of coefficients, shift lines of integration to pick up polar main terms, apply the functional equation and change variables so work with the dual, and then get some collection of error terms and error integrals.

It happens to be that it’s much easier to work with a $k$-Riesz smoothed approximation. That is, if $$
\phi(s) = \sum_{n \geq 1} \frac{a(n)}{\lambda_n^s}
$$
is our Dirichlet series, and we are interested in the partial sums $$
A_0(s) = \sum_{\lambda_n \leq X} a(n),
$$
then it happens to be easier to work with the smoothed approximations $$
A_k(X) = \frac{1}{\Gamma(k+1)}\sum_{\lambda_n \leq X} a(n) (X – \lambda_n)^k a(n),
$$
and to somehow combine several of these smoothed sums together.

This smoothed sum is recognizable as $$
A_k(X) =
\frac{1}{2\pi i}\int_{c – i\infty}^{c + i\infty} \phi(s)
\frac{\Gamma(s)}{\Gamma(s + k + 1)} X^{s + k}ds
$$
for $c$ somewhere in the half-plane of convergence of the Dirichlet series. As $k$ gets large, these integrals become better behaved. In application, one takes $k$ sufficiently large to guarantee desired convergence properties.

The process of taking several of these smoothed approximations for large $k$ together, studying them through basic functional equation methods, and combinatorially combining these smoothed approximations via finite differencing to get good estimates for the sharp sum $A_0(s)$ is roughly what I think of as “Landau’s Method”.

Application and shape of the error

In our paper, as we apply Landau’s method, it becomes necessary to understand certain bounds coming from the dual Dirichlet series $$
\psi(s) = \sum_{n \geq 1} \frac{b(n)}{\mu_n^s}.
$$
Specifically, it works out that the (combinatorially finite differenced) between the $k$-smoothed sum $A_k(X)$ and its $k$-smoothed main term $S_k(X)$ can be written as $$
\Delta_y^k [A_k(X) – S_k(X)] = \sum_{n \geq 1}
\frac{b(n)}{\mu_n^{\delta + k}} \Delta_y^k I_k(\mu_n X),\tag{1}
$$
where $\Delta_y^k$ is a finite differencing operator that we should think of as a sum of several shifts of its input function.

More precisely, $\Delta_y F(X) := F(X + y) – F(X)$, and iterating gives $$
\Delta_y^k F(X) = \sum_{j = 0}^k (-1)^{k – j} {k \choose j} F(X + jy).
$$
The $I_k(\cdot)$ term on the right of $(1)$ is an inverse Mellin transform $$
I_k(t) = \frac{1}{2 \pi i} \int_{c – i\infty}^{c + i\infty}
\frac{\Gamma(\delta – s)}{\Gamma(k + 1 + \delta – s)}
\frac{\Delta(s)}{\Delta(\delta – s)} t^{\delta + k – s} ds.
$$
Good control for this inverse Mellin transform yields good control of the error for the overall approximation. Via the method of finite differencing, there are two basic choices: either bound $I_k(t)$ directly, or understand bounds for $(\mu_n y)^k I_k^{(k)}(t)$ for $t \approx \mu_n X$. Here, $I_k^{(k)}(t)$ means the $k$th derivative of $I_k(t)$.

Large input errors

In the classical application (as in the paper of CN), one worries about this asymptotic mostly as $t \to \infty$. In this region, $I_k(t)$ can be well-approximated by a $J$-Bessel function, which is sufficiently well understood in large argument to give good bounds. Similarly, $I_k^{(k)}(t)$ can be contour-shifted in a way that still ends up being well-approximated by $J$-Bessel functions.

The shape of the resulting bounds end up being that $\Delta_y^k I_k(\mu_n X)$ is bounded by either

  • $(\mu_n X)^{\alpha + k(1 – \frac{1}{2A})}$, where $A$ is a fixed parameter that isn’t worth describing fully, and $\alpha$ is a bound coming from the direct bound of $I_k(t)$, or
  • $(\mu_n y)^k (\mu_n X)^\beta$, where $\beta$ is a bound coming from bounding $I_k^{(k)}(t)$.

In both, there is a certain $k$-dependence that comes from the $k$-th Riesz smoothing factors, either directly (from $(\mu_n y)^k$), or via its corresponding inverse Mellin transform (in the bound from $I_k(t)$). But these are the only aspects that depend on $k$.

At this point in the classical argument, one determines when one bound is better than the other, and this happens to be something that can be done exactly, and (surprisingly) independently of $k$. Using this pair of bounds and examining what comes out the other side gives the original result.

Small input errors

In our application, we also worry about asymptotic as $t \to 0$. While it may still be true that $I_k$ can be approximated by a $J$-Bessel function, the “well-known” asymptotics for the $J$-Bessel function behave substantially worse for small argument. Thus different methods are necessary.

It turns out that $I_k$ can be approximated in a relatively trivial way for $t \leq 1$, so the only remaining hurdle is $I_k^{(k)}(t)$ as $t \to 0$.

We’ve proved a variety of different bounds that hold in slightly different circumstances. And for each sort of bound, the next steps would be the same as before: determine when each bound is better, bound by absolute values, sum together, and then choose the various parameters to best shape the final result.

But unlike before, the boundary between the regions where $I_k$ is best bounded directly or bounded via $I_k^{(k)}$ depends on $k$. Aside from choosing $k$ sufficiently large for convergence properties (which relate to the locations of poles and growth properties of the Dirichlet series and gamma factors), any sufficiently large $k$ would suffice.

Limiting behavior gives a heuristic region

After I step away from this paper and argument for a while and come back, I wonder about the right way to choose the balancing error. That is, I rework when to use bounds coming from studying $I_k(t)$ directly vs bounds coming from studying $I_k^{(k)}(t)$.

But it turns out that there is always a reasonable heuristic choice. Further, this heuristic gives the same choice of balancing as in the case when $t \to \infty$ (although this is not the source of the heuristic).

Making these bounds will still give bounds for $\Delta_y^k I_k(\mu_n X)$ of shape

  • $(\mu_n X)^{\alpha + k(1 – \frac{1}{2A})}$, where $A$ is a fixed parameter that isn’t worth describing fully, and $\alpha$ is a bound coming from the direct bound of $I_k(t)$, or
  • $(\mu_n y)^k (\mu_n X)^\beta$, where $\beta$ is a bound coming from bounding $I_k^{(k)}(t)$.

The actual bounds for $\alpha$ and $\beta$ will differ between the case of small $\mu_n X$ and large $\mu_n X$ ($J$-Bessel asymptotics for large, different contour shifting analysis for small), but in both cases it turns out that $\alpha$ and $\beta$ are independent of $k$.

This is relatively easy to see when bounding $I_k^{(k)}(t)$, as repeatedly differentiating under the integral shows essentially that $$
I_k^{(k)}(t) =
\frac{1}{2\pi i}
\int \frac{\Delta(s)}{(\delta – s)\Delta(\delta – s)}
t^{\delta – s} ds.
$$
(I’ll note that the contour does vary with $k$ in a certain way that doesn’t affect the shape of the result for $t \to 0$).

When balancing the error terms $(\mu_n X)^{\alpha + k(1 – \frac{1}{2A})}$ and $(\mu_n y)^k (\mu_n X)^\beta$, the heuristic comes from taking arbitrarily large $k$. As $k \to \infty$, the point where the two error terms balance is independent of $\alpha$ and $\beta$.

This reasoning applies to the case when $\mu_n X \to \infty$ as well, and gives the same point. Coincidentally, the actual $\alpha$ and $\beta$ values we proved for $\mu_n X \to \infty$ perfectly cancel in practice, so this limiting argument is not necessary — but it does still apply!

I suppose it might be possible to add another parameter to tune in the final result — a parameter measuring deviation from the heuristic, that can be refined for any particular error bound in a region of particular interest.

But we haven’t done that.

In fact, we were slightly lossy in how we bounded $I_k^{(k)}(t)$ as $t \to 0$, and (for complicated reasons that I’ll probably also forget and reprove to myself later) the heuristic choice assuming $k \sim \infty$ and our slighly lossy bound introduce the same order of imprecision to the final result.

More coming soon

We’re updating our preprint and will have that up soon. But as I’ve been thinking about this a lot recently, I realize there are a few other things I should note down. I intend to write more on this in the short future.

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Slides from a talk at AIM

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.

Here are the slides for my talk.

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.

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Slides from a talk on computing Maass forms

Yesterday, I gave a talk on various aspects of computing Maass cuspforms at Rutgers.

Here are the slides for my talk.

Unlike most other talks that I’ve given, this doesn’t center on past results that I’ve proved. Instead, this is a description of an ongoing project to figure out how to rigorously compute many Maass forms, implement this efficiently in code, and add this data to the LMFDB.

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Talk on computing Maass forms

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.

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The current cover art for the Proceedings of the Royal Society

The current issue of the Proceedings of the Royal Society A1 features cover artwork made by Vikas Krishnamurthy, Miles Wheeler, Darren Crowdy, Adrian Constantin, and me.

Cover of Proceedings of the Royal Society AA 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.

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phase_mag_plot: a sage package for plotting complex functions

Inspired by conversations with Elias Wegert and Frank Farris at the Illustrating Mathematics semester program at ICERM last year, I wrote several plotting libraries for complex plotting. I wrote them with the intention of plotting modular forms in a variety of ways, leading to my talk at Bowdoin in November 2019 and my first post on the CS arxiv.1

I’ve gotten several requests to make these plotting libraries available, and so I’ve made davidlowryduda/phase_mag_plot available on github as a sage library. See the github page and README for examples and up-to-date information.

This version is capable of producing contour-type plots of complex functions.

A plot of x^2(x-3)(x-3i) with magnitude-type contoursA plot of x^2(x-3)(x-3i) with tile-type contours

This does not include any colormap capability yet, as that is a substantially more involved2 process. But at some point in the future, I intend to look at revisiting the complex plotting within sage itself, perhaps updating it to allow plots of this nature.

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Notes from a talk at Dartmouth on the Fibonacci zeta function

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.

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Pictures of equidistribution – the line

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.

(more…)

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