MixedMath - explorations in math and programing

Initial thoughts on visualizing number fields

David Lowry-Duda — Fri, 22 Jul 2022 03:14:15 +0000

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Computing coset representatives for quotients of congruence subgroups

David Lowry-Duda — Wed, 20 Jul 2022 03:14:15 +0000

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Note Series on Zeros of General Dirichlet Series

David Lowry-Duda — Sat, 09 Jul 2022 03:14:15 +0000

I wrote a series of notes on some aspects of the theoretical behavior and zeros of Dirichlet series in the extended Selberg Class. There are a few different ways of extending the Selberg Class, but here I mean Dirichlet series \begin{equation*} L(s) = \sum_{n \geq 1} \frac{a(n)}{n^s} \end{equation*} with a functional equation of the shape $s \mapsto 1 - s$, satisfying

A Ramanujan–Petersson bound on average, meaning that $\sum_{n \leq N} \lvert a(n) \rvert^2 \ll N^{1 + \epsilon}$ for any $\epsilon > 0$.
$L(s)$ has analytic continuation to $\mathbb{C}$ to an entire function of finite order.
$L(s)$ satisfies a functional equation of the typical self-dual shape.

There is no assumption of an Euler product.

Although I write these notes for general Dirichlet series in the extended Selberg class, I was really thinking about Dirichlet series associated to half-integral weight modular forms.

Links and Summaries of each Note

The first note sets the stage, defines the relevant series, and establishes fundamental results to be used later. Jensen's inequality and Jensen's theorem are given, as are generic convexity bounds for these Dirichlet series.

The first note also contains a proof of a new fact to me:¹ ¹but only new to me. I based my presentation of this fact on notes from Hardy from a century ago. if a Dirichlet series has a zero in its domain of absolute convergence, then it has infinitely many, and these zeros are almost periodic.
The second note is relatively short and shows that these Dirichlet series are in fact entire of order $1$. Then it establishes weak zero-counting results based only on this order of growth.

These are foundational ideas, and in essence are no different than analysis for $\zeta(s)$ or typical $L$-functions in the Selberg class.
The third note describes a theorem of Potter from 1940, proving Lindelöf-on-average (in the $t$-aspect) on certain vertical lines, depending on the degree. This suggests that for Dirichlet series associated to half-integral weight modular forms, the Lindelöf Hypothesis might be true even though the Riemann Hypothesis is false.
The fourth note proves that one hundred percent of zeros of Dirichlet series associated to half-integral weight modular forms lie within $\epsilon$ of the critical line, for any $\epsilon > 0$.

For these proofs, the standard proofs that I've seen elsewhere for Selberg $L$-functions don't quite apply. Fundamentally, this is because zeros are counted using some form of the argument principle, and for Selberg $L$-functions with Euler product, the logarithmic derivative of the $L$-functions is both easier to understand (because of the Euler product) and gives convenient access to sum up changes in the argument. But in practice the actual methods I use were mostly lightly modified from the vast, extensive literature on various techniques applied to study $\zeta(s)$ (before better techniques arose using the Euler product in more sophisticated ways).

I note that I do not know how to prove a lower bound for the percentage of zeros lying directly on the critical line for series without an Euler product. It is possible to apply a generalized form of the classical argument of Hardy to show that there are infinitely many zeros on the line, but I haven't managed to modify any of the various results for lower bounds.

I will also note that with Thomas Hulse, Mehmet Kiral, and Li-Mei Lim, I've computed many examples of zeros of half-integral weight forms and I conjecture that 100 percent of their zeros lie directly on the critical line. (But there are many, many zeros not on the critical line). I described some of those computations in this talk.

Zeros of Dirichlet Series IV

David Lowry-Duda — Fri, 08 Jul 2022 03:14:15 +0000

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Zeros of Dirichlet Series III

David Lowry-Duda — Thu, 07 Jul 2022 03:14:15 +0000

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Visualizations for Quanta's 'What is the Langlands Program?'

David Lowry-Duda — Wed, 01 Jun 2022 03:14:15 +0000

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Now page

David Lowry-Duda — Tue, 10 May 2022 03:14:15 +0000

What I'm doing Now

Last updated 3 September 2022.

This is a now page. It is a written version of what I might say if we met in person and you asked me what I'm up to. The date at the top is important: it helps the reader determine if this is a now page or a then page.

Research Travel

I will be near Berkeley during the week of September 12th.
I will be at Maine-Quebec on October 15-16.

Other than that, I'm typically in either Boston or Providence.

Research

I'm currently actively working on several projects.

I'm working to develop methods to explicitly and rigorously compute Maass forms. I'm currently implementing a rigorous form of Hejhal's algorithm for weight $0$ Maass forms in arb.
At the same time, I'm currently writing down interrelated things about Maass forms, including implementation notes for my computational work, the theory behind rigorous evaluation, and work towards an explicit and computable trace formula for forms of nontrivial character on non-squarefree level.
With my frequent collaborators Chan Ieong Kuan and Alex Walker, I'm investigating discrete reduction problems for the divisor function. We will release a preprint for this soon.
I supervised a project with some students at PROMYS. We're writing down several of their observations.

Teaching

I'm not currently teaching.

Slides from a talk on Improved Bounds for Number Fields

David Lowry-Duda — Mon, 09 May 2022 03:14:15 +0000

At a meeting of the Algebraic Geometry, Number Theory, and Computation Simons Collaboration, I gave a short talk surveying the results and ideas of Improved bounds on number fields of small degree, joint work with Anderson, Gafni, Hughes, Lemke Oliver, Thorne, Wang, and Zhang.

In many ways this is a follow-up to an earlier talk I gave to the Simons Collaboration about work towards a quantitative form of Hilbert's irreducibility theorem. Both of these results grew out of an AIM workshop.

The slides for this talk are available here.

Simplified proofs and reasoning in "Improved Bounds on Number Fields of Small Degree"

David Lowry-Duda — Sat, 30 Apr 2022 03:14:15 +0000

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Paper: Counting number fields of small degree

David Lowry-Duda — Thu, 28 Apr 2022 03:14:15 +0000

Counting number fields of small degree

Recently, my collaborators Theresa C. Anderson, Ayla Gafni, Kevin Hughes, Robert J. Lemke Oliver, Frank Thorne, Jiuya Wang, Ruixiang Zhang, and I uploaded a preprint to the arxiv called "Improved bounds on number fields of small degree". This collaboration is a continuation¹ ¹though with a few different cast members of our previous work on quantitative Hilbert irreducibility, which will appear in IMRN.

In this paper, we improve the upper bound due to Schmidt for estimates on the number of number fields of degree $6 \leq n \leq 94$. Actually, we improve on Schmidt for all $n \geq 6$, but for $n \geq 95$ Lemke Oliver and Thorne have different, better bounds.

Schmidt proved the following.

For $n \geq 6$, there are $\ll X^{(n+2)/4}$ number fields of degree $n$ and having discriminant bounded by $X$.

We prove a polynomial improvement that decays with the degree.

For $n \geq 6$, there are \begin{equation*} \ll_\epsilon X^{\frac{n + 2}{4} - \frac{1}{4n - 4} + \epsilon} \end{equation*} number fields of degree $n$ and having discriminant bounded by $X$.

Towards the end of this project, we learned that Bhargava, Shankar, and Wang were also producing improvements over Schmidt in this range. On the same day that we posted our paper to the arxiv, they posted their paper, in which they prove the following.

For $n \geq 6$, there are \begin{equation*} \ll_\epsilon X^{\frac{n + 2}{4} - \frac{1}{2n - 2} + \frac{1}{2^{2g}(2n-2)} + \epsilon} \end{equation*} number fields of degree $n$ and having discriminant bounded by $X$, where $g = \lfloor \frac{n-1}{2} \rfloor$.

In both our work and in BSW, the broad strategy is based on Schmidt's approach. For a monic polynomial \begin{equation*} f(x) = x^n + c_1 x^{n-1} + \cdots + c_n, \end{equation*} we define the height $H(f)$ to be \begin{equation*} H(f) := \max( \lvert c_i \rvert^{1/i} ). \end{equation*} Then Schmidt showed that to count number fields of discriminant up to $X$, it suffices to count polynomials of height roughly up to $X^{1/(2n - 2)}$.

The challenge is that most of these polynomials cut out number fields of discriminant much larger than $X$. The challenge is then to count relevant polynomials and to identify irrelevant polynomials.

Remarkably, the broad strategy in out work and in BSW for identifying irrelevant polynomials is similar. For a prototypical polynomial $f$ of degree $n$ and of height $X^{1/(2n-2)}$, we should expect the discriminant of $f$ to be approximately $X^{n/2}$. We should also expect the field cut out by $f$ to have discriminant roughly this size. Recalling that we are counting number fields of discriminant only up to $X$, this means that a relevant polynomial of this height must be exceptional in one of two ways:

either the discriminant of $f$ is unusually small, or
the discriminant of the number field cut out by $f$ is much smaller than the discriminant of $f$.

In both out work and in BSW, those $f$ with unusually small discriminant are bounded straightforwardly and lossily.

The heart of the argument is in the latter case. Here, the ratio of the two discriminants is the square of the index $[\mathcal{O}_K : \mathbb{Z}[\alpha]]$, where $\alpha$ is a root of $f$. Thus we bound the number of polynomials whose discriminants have large square divisors.

In establishing bounds for polynomials with particularly squarefull discriminants that our ideas and those in BSW significantly diverge.

In our work, we study the problem locally. That is, we study the behavior of $\psi_{p^{2k}}$, the characteristic function for monic polynomials of degree $n$ over $\mathbb{Z}/p^{2k}\mathbb{Z}$ having discriminant congruent to $0 \bmod p^{2k}$. As in our work on quantitative Hilbert irreducibility, we translate this problem into a sieve problem with local weights coming from Fourier transforms $\widehat{\psi_{p^{2k}}}$ after passing through Poisson summation, and we study the Fourier transforms using a variety of somewhat ad-hoc techniques.

In BSW, they reason differently. They use recent explicit quantitative Hilbert irreducibility work from Castillo and Dietmann to replace the fundamental underlying sieve. To do this, they translate the task of counting relevant polynomials into the task of counting distinguished points in spaces of $n \times n$ symmetric matrices — and then show that Castillo and Dietmann's work bounds these points.

Even though the number field count in BSW is stronger than our number field count, we think that our methods and ideas will have other applications. Further, we've noticed remarkable interactions between local Fourier analysis and discriminants of polynomials.

MixedMath - explorations in math and programing

Initial thoughts on visualizing number fields

Computing coset representatives for quotients of congruence subgroups

Note Series on Zeros of General Dirichlet Series

Links and Summaries of each Note

Zeros of Dirichlet Series IV

Zeros of Dirichlet Series III

Visualizations for Quanta's 'What is the Langlands Program?'

Now page

What I'm doing Now

Research Travel

Research

Teaching

Slides from a talk on Improved Bounds for Number Fields

Simplified proofs and reasoning in "Improved Bounds on Number Fields of Small Degree"

Paper: Counting number fields of small degree

Counting number fields of small degree

See also