We had a JetBlue flight for like a minute maybe 15 years ago but it didn't last long.

Shouldn't the argument where you prove it first for collections of length 2^k by induction on k, and then in general by padding out a sequence of length n by appending 2^k - n copies of the average, just go over almost directly?

I don't get it. Why bother with AI subjects when you could skip directly to having the AI write the paper with fabricated data instead.

Best place to wive it wealthily

Also Sunday evening, Monday morning through evening, Tuesday.....

The notes for the Mathcamp course are here, for anyone interested:
tinyurl.com/p6bvnrwp
In 2 weeks / 10 lectures I made it to the end of section 3, i.e., the level one stuff.

(36/36)

Alright, let me stop here. Thanks for reading to the end. The moral, if there is one, is: thinking about how to explain interesting things to smart high school students is great, and if you have the opportunity, you should do it.

Less surprisingly, the LLMs were quite helpful with proofreading. Gemini especially caught some typos that I'm very glad it caught.

LLMs do get credit for one trick that made it into the paper, the idea of multiplying by (b/d)^2 in the proof of Lemma 4.9 (though what the LLM suggested was more elaborate and the version in the paper is streamlined).

It can often be useful to try to understand why some argument is wrong, and the points on which the LLM was most wrong were exactly the key points. For example the one multiplier system formula I use gives its values on a particular level 24 subgroup.

They tried to convince me that the multiplier system for eta(z) is trivial on Gamma(24) (false). And later it gave me a beautiful but totally wrong proof of Theorem 4.1 using Gauss sums.

But I would say that each of these mistakes stimulated my own thinking in productive ways.

... maybe in some dispersed fashion, and LLMs would explain them to me.

What happened was more interesting. I don't think LLMs contributed any ideas to the argument, and they contributed several anti-ideas. They were hard to dissuade from the idea that T and L_N generate Gamma_1(N), for example.

I want to talk a little bit about my use of LLMs while writing this article. While working on this, I had running conversations going with with Gemini 2.5 Pro and GPT-o3 --- mostly because I expected at the outset that modern treatments of Newman's theorem would already exist...

To prove the latter observation, we still require one formula involving the multiplier system for eta. But this formula is elementary in the sense that no Dedekind sums are required either in its statement or its proof. So this is a "Dedekind sum free" proof of the theorem.

The second observation is that any eta-quotient of integer weight is necessarily modular for some congruence subgroup (with M possibly very large). Newman's theorem follows.

But these two matrices do NOT generate Gamma_1(N), so why should this condition be sufficient?

One basic observation in today's note is that Gamma_1(N) is generated by T and L_N together with any congruence subgroup (i.e. containing matrices congruent to I mod M for some M).

Now, the two conditions in Newman's theorem (the two divisibilities by 24) are precisely the necessary condition that an eta-quotient transform correctly under the two matrices T and L_N:
[ 1 1 ]
[ 0 1 ]
and
[ 1 0 ]
[ N 1 ] .

... to deduce that the multiplier system is trivial on such elements. It all works out, but doesn't seem to explain anything.

Gamma_(1)(N) is the subgroup of matrices whose lower-left entry is divisible by N, and diagonal entries are 1 modulo N.

Plug an element of Gamma_1(N) into the multiplier system for the eta-quotient. Then cleverly apply various baroque Dedekind sum identities and congruences...