MixedMath - Recent Comments

DLD on Paper: The Fibonacci Zeta Function and Continuation

Wed, 05 Feb 2025 03:14:15 +0000

We've now also uploaded our sequel paper to the arxiv. In this paper, we focus on modular forms.

DLD on Odd Fibonacci Zeta Function

Wed, 05 Feb 2025 03:14:15 +0000

We've now also uploaded our sequel paper to the arxiv. In this paper, we focus on modular forms.

We promised this paper in the first paper.

davidlowryduda on An (updated) brief note on cryptography

Mon, 30 Dec 2024 03:14:15 +0000

I should thank some readers for pointing out that my previous code was old and no longer ran. Cheers, and happy new year.

davidlowryduda on A brief notebook on cryptography

Mon, 30 Dec 2024 03:14:15 +0000

@anon You're right! This is very python2. I'll post an updated version right now. It's now here.

anon on A brief notebook on cryptography

Mon, 30 Dec 2024 03:14:15 +0000

I think this doesn't work with python3.10 anymore? I'm not familiar enough with OO languages to figure this out. Can you provide a solution?

DLD on Odd Fibonacci Zeta Function

Wed, 04 Dec 2024 03:14:15 +0000

I experiment with mirroring some commentary from Mastodon and here. This is from @davidlowryduda@mathstodon.

The Dirichlet series $Z(s) = \sum F(n)^{-s}$ where $F(n)$ is the $n$th Fibonacci number has meromorphic continuation. And it looks pretty cool.

DLD on Murmurations in Maass forms

Fri, 25 Oct 2024 03:14:15 +0000

As part of the program in Mathematics and Machine Learning at Harvard's CMSA, we looked further at what machine learning (both unsupervised and supervised) techniques have to say about Maass forms, murmurations style.

Despite the data being transcendental and approximate, the machines can say a lot! We're going to have a preliminary talk about this during the closing workshop of the program in the coming week. (These talks are recorded — you can see a preview of our preprint in Edgar Costa's talk, and a preview of a different preprint in my talk).

David Lowry-Duda on Technical report on machine learning experiments for the Möbius function

Thu, 24 Oct 2024 03:14:15 +0000

All the code to run the experiments is here or in the Int2Int github repository I linked to. It's not as hard to run as it might look!

I don't know what you mean about primes. But if you set up an experiment along these lines, I'd be interested to hear about it.

SH on Technical report on machine learning experiments for the Möbius function

Wed, 23 Oct 2024 03:14:15 +0000

Can you share the GPU code? Can you predict primes?

DLD on Explicit equations for cubic surfaces

Thu, 26 Sep 2024 03:14:15 +0000

Excellent observation! At first I thought a lot was lost by choosing the points to be $6$ points in $\mathbb{P}^2(\mathbb{F}_q)$ instead of consisting of Galois orbits. But actually passing to (at most) sextic extensions and doing the same analysis barely loses anything.

The analysis is similar.

This is an interesting heuristic. I wonder: can something like this be made rigorous?

anon on Explicit equations for cubic surfaces

Thu, 26 Sep 2024 03:14:15 +0000

Consider an analogous problem over $\mathbb{F}_q$. We first count the number of ''cubic surfaces'' here, i.e. homogeneous cubic $4$-variable polynomials over $\mathbb{F}/\mathbb{F}^\times$.

There are $q^{20} - 1$ homogenous, four-variable cubic polynomials with coefficients in $\mathbb{F}$. The $20$ is a ''stars and bars game'' analysis. There are $3$ powers that we insert into $4$ regions, which can be done in ${4 + 3 - 1 \choose 3} = 20$ ways. Accounting for $\mathbb{F}^\times$ invariance, there are $M := (q^{20} - 1)/(q - 1) > q^{19}$ equivalence classes.

Now suppose we want to count the number of choices of any $6$ rational points. The number of points in $\mathbb{P}^2(\mathbb{F}_q)$ is $n := (q^3 - 1)/(q-1) = q^2 + q + 1$. There are up to $N := {n \choose 6} \lesssim q^{12}$ many choices of $6$ points to construct blowups.

Thus the probability that a random rational cubic surfaces comes "from blowing up $6$ points" in $\mathbb{P}^2(\mathbb{F}_q)$ is $< q^{-7}$, quite small.

DLD on Murmurations in Maass forms

Thu, 05 Sep 2024 03:14:15 +0000

We've now proved a version of these murmurations. See this paper on the arxiv for more.

There is a miraculus aspect: the underlying murmuration function is the same as the function for modular murmurations in the weight aspect. I have no satisfying explanation for this.

DLD on Maass forms in the LMFDB

Wed, 05 Jun 2024 03:14:15 +0000

I experiment with mirroring some commentary from Mastodon and here. This is from @davidlowryduda@mathstodon.

Rigorous Maass forms are now on the (beta) of the LMFDB! Go check them out.

https://beta.lmfdb.org/ModularForm/GL2/Q/Maass/

DLD on Another year, another TeXLive reinstallation

Mon, 15 Apr 2024 03:14:15 +0000

I experiment with mirroring some commentary from Mastodon and here. This is from @davidlowryduda@mathstodon.

Once a year, sometime shortly after March, I try to update something with tlmgr (the texlive manager) and it flails about. It tells me that I'm silly for not noticing that TeXLive has advanced another year, and that I should get with the times and update too.

Then I spend 30 minutes updating my texlive.

And then afterwards I can install the random latex package I needed.

Today is that day for me. And the package was extdash.

I do essentially the same steps every year. This year I put these steps in a public place.

DLD on FLT3 at LftCM2024

Tue, 02 Apr 2024 03:14:15 +0000

Thank you to Edward van de Meent and Riccardo Brasca for comments and pointing out errors in earlier versions of this post.

DLD on FLT3 at LftCM2024

Tue, 02 Apr 2024 03:14:15 +0000

I experiment with mirroring some commentary from Mastodon and here. This is from @davidlowryduda@mathstodon.

I attended Lean for the Curious Mathematician 2024 at CIRM. Along with Riccardo Brasca, Sanyam Gupta, Omar Haddad, Lorenzo Luccioli, Pietro Monticone, Alexis Saurin, and Florent Schaffhauser, we proved the $n=3$ case of Fermat's Last Theorem in a good way in Lean.

I wrote a bit more about this experience.

DLD on Zeros of Dirichlet Series IV

Wed, 06 Mar 2024 03:14:15 +0000

Thank you! I've fixed the typo.

I haven't published these yet. I'm going to try to put this in a publishable form in the next couple of months.

DLD on Quanta on Murmurations

Tue, 05 Mar 2024 03:14:15 +0000

I experiment with mirroring some commentary from Mastodon and here. This is from @davidlowryduda@mathstodon.

Quanta wrote an article on murmurations that mentions the work of my collaborators Bober, Booker, Lee, and I.

We're not the focus, but we're there - and that's pretty cool. Check it out.

I hope to announce another paper in the same direction in the next couple of weeks, too.

Chris on Zeros of Dirichlet Series IV

Tue, 05 Mar 2024 03:14:15 +0000

Have you published this yet? There is a typo in the second display equation. It should have a $\Lambda$ instead of a $\lambda$.

PJ on Paper: Congruent number triangles with the same hypotenuse

Thu, 15 Feb 2024 03:14:15 +0000

@RB can't find more. You already found them all

DLD on Paper: Towards a Classification of Isolated $j$-invariants

Wed, 14 Feb 2024 03:14:15 +0000

I'll give a small update as a comment. Our paper has been accepted and will appear in Mathematics of Computation shortly. (And Math Comp was very fast and forthright with their review, which is nice).

David Lowry-Duda on Examining Excess in the Schmidt Bound

Tue, 13 Feb 2024 03:14:15 +0000

I will also note that I hadn't expected this to be so cut and dry, and I ran a separate experiment that sampled randomly from the space of polynomials and went in that direction.

The setup and conclusion are more complicated. But the conclusion is the same. So I chose to not document or include it.

David Lowry-Duda on Examining Excess in the Schmidt Bound

Tue, 13 Feb 2024 03:14:15 +0000

I experiment with mirroring some commentary from Mastodon and here. This is from @davidlowryduda@mathstodon.xyz.

I study excess in the Schmidt bound for number fields. Schmidt estimated that there are at most $X^{\frac{n + 2}{4}}$ number fields of degree $n$ with discriminant up to $X$. But we expect the real number is really of the form $cX$ for some constant $c$.

To do this, Schmidt gives a region of polynomials to count, where some polynomial is guaranteed to generate each number field of discriminant up to $X$. There are two sorts of error there: counting polynomials that yield number fields with discriminant larger than $X$, and sets of polynomials that all count the same number field.

Which error is larger?

For heuristic reasons, we should expect counting polynomials that are too large to be a major source of overcounting. So perhaps a meaningful question would be: does counting the same number field multiple times account for much of the error at all?

In this note I describe an experiment and show that no, repeated counting isn't a significant factor.

DLD on Plaintext Email

Wed, 24 Jan 2024 03:14:15 +0000

I use K-9 on mobile, and a variety of clients on various machines.

The mobile app for gmail cannot send plaintext. By default, the web interface doesn't — but it can.

Plaintext is worse for marketing. You can't hide links or track image loading for additional analytics, so it is maybe no surprise that ad-based companies default to HTML email.

gmailer on Plaintext Email

Tue, 23 Jan 2024 03:14:15 +0000

What email do you use? gmail doesn't use plaintext?