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# Draft for seminar program

Each lecture hopefully requires only 1 hour.

Part 1: Introduction - 3 lectures - [L]

Part 2: Generalities of Nilsequences - 2 or 3 lectures - [Q]

Part 3: Equidistribution - 2 lectures - [Book] and [P]

Part 4: MN(s) - 1 lecture - [M]

Part 5: GI(s) - ?? lectures - [GTZ]

Part 6: Back to linear equations - ?? lectures - [L]

[L] = Linear equations in primes, [Q] = The quantitative behaviour of polynomial sequences, [Book] = Higher Order Fourier Analysis, [M] = The Mobius function is strongly orthogonal to nilsequences, [GTZ] = Green-Tao-Ziegler.

### Part 1: Main results of linear equations in primes

In this paper, the main results are proven assuming conditions called GI(s) and MN(s). These two conditions were later confirmed by Green-Tao and Green-Tao-Ziegler.

References: the linear equation paper (98 pages) by Green and Tao

#### Lecture 1

-§§ 1, 4 of the paper.

-Introduce enough notions to explain how the results announced in Introduction of the linear equation paper are related to each other.

-There are linear algebra tricks involved; though they are elementary in nature, it would be nice to cover them to make sure we are on the same page.

#### Lecture 2

- Statement of the conjectures GI(s) and MN(s) from §8 of the linear equation paper.

- To state this, introduce the Gowers norm (linear equation paper Appendix B) and the notion of nilsequences following the linear equation paper §8. This latest Bourbaki Seminar article (2020) also gives a compact summary.

#### Lecture 3

Try your best to explain how the main result can be proven under GI(s) and MN(s). At this stage, a rough roadmap would suffice.

### Part 2: Generalities on simply connected nilpotent Lie groups and polynomial sequences

Such Lie groups and sequences appear in the statements of GI(s) and MN(s).

References: Green's videos (a YouTube playlist; the list in Green's page is also useful http://people.maths.ox.ac.uk/greenbj/videos.html ),

the polynomial orbits paper (76 pages) by Green and Tao.

How to get prepared:

Watch video 2 and 3 of Green's 6 lectures.

Read §§ 1, 2 and 6 of the polynomial orbits paper.

#### Lecture 1

Explain the following notions perhaps with illustrating examples:

##### - simply connected nilpotent Lie groups/algebras.

The category of connected nilpotent Lie groups is equivalent to the category of nilpotent Lie algebras via the functor $G\mapsto \mathfrak g := \mathrm{Lie}(G)$. (The essential part of this equivalence is called Lie's third theorem. See also this post of mine (in Japanese).)

##### - filtered groups in the sense of Green-Tao

See Green's video 2/6

https://youtu.be/27Ljd_M--Qw?t=1285

and Introduction to the polynomial orbits paper.

##### - Lattices and Malcev bases of simply connected nilpotent Lie groups

See Green's video 2/6 44:00-47:00

https://youtu.be/27Ljd_M--Qw?t=2626

Green's video 2/6 56:30--1:03:00

https://youtu.be/27Ljd_M--Qw?t=3390

and §2 of the polynomial orbits paper.

#### Lecture 2

##### - define the notion of polynomial maps $H\to G$ between filtered groups and in particular polynomial sequences $\mathbb Z \to G$; equivalent formulations of these notions.

See Green's video 3/6 11:00-21:00

https://youtu.be/u4inkf0VJAU?t=669

and §6 of the polynomial orbits paper.

Section 1.6 of Tao's Higher Order Fourier Analysis is also useful.

#### Lecture 3 (optional)

##### - Present a proof of the converse to GI(s) (the easier implication).

Since it is not logically needed, this lecture may be skipped. However, the proof in Green's video 3/6 56:00

https://youtu.be/u4inkf0VJAU?t=3379

contains nice arguments using induction on $s$. This proof also shows how the notion of general polynomial sequences is useful when one is only interested in the linear sequence $n\mapsto g^n x$.

You might also want to refer to Proposition 8.2 of the paper "Linear equations in primes," which refers to Proposition 12.6 of the paper "An inverse theorem for the Gowers $U^3(G)$ norm."

https://doi.org/10.1017/S0013091505000325

### Part 3: Equidistribution of polynomial sequences

This seems to be an important ingredient for the proof of both conjectures.

#### Optional activity

One could read pp. 2--27 of Tao's book "Higher order Fourier analysis" to familiarize themselves with the equidistribution phenomenon of sequences defined by polynomials $f\in \mathbb R [X]$: $\mathbb Z \xrightarrow{n\mapsto f(n) } \mathbb R \longrightarrow \mathbb R /\mathbb Z$ which is classical. We could as well have a homework session discussing this classical case.

#### How to get prepared:

Watch this video of Green's (= Lecture 4/6). It explains a rough idea of what the topic is about. See this post
https://motivichomotopy.hatenablog.jp/entry/2021/03/02/123330
for what is done in this video.

Have a quick look through The quantitative behaviour of polynomial orbits on nilmanifolds by Green and Tao. In doing so, you could read the introduction first and take a comprehension test here

Note that this paper contains errors in §8. See this erratum (the arXiv version is more detailed: https://arxiv.org/abs/1311.6170 ). In Erratum, the authors describe how to establish the results in the parallel context of multivariate polynomial sequences $\mathbb Z ^t \to G$.

#### Lecture 1

##### - main results.

Introduce enough materials from the paper so that you can  state its main results (Theorems 2.9 and 1.19) and explain how you can (or cannot) appreciate their beauty/importance.

##### - from Theorem 2.9 to Theorem 1.19.

Make a comment on how Theorem 2.9 can be used to prove Theorem 1.19.

We don't have to know the proof of Theorem 2.9. It's too technical.

#### Lecture 2

##### - Explain some technical contents of the paper https://doi.org/10.4007/annals.2012.175.2.2and the Erratum. https://arxiv.org/abs/1311.6170

In particular, Lemma 2.1 from Erratum, Proposition 2.3 from Erratum, Proposition 8.4 from the paper and Theorem 8.6 from the paper (whose proof is completed in Erratum) turn out to be useful in the proof of MN(s). The case $N_1=\cdots =N_t=N$ suffices. See also this post of mine. https://motivichomotopy.hatenablog.jp/entry/2021/01/12/005343

### Part 4: The condition MN(s)

This was proven in the paper titled "The Mobius function is strongly orthogonal to nilsequences" (26 pages) by Green-Tao. It was published directly after the polynomial orbits paper in the same volume.

Good news is that the main content of this paper is completed within the first 13 pages.

### Part 5: The codition GI(s)

Main reference is this paper by Green-Tao-Ziegler.(142 pages...)

You might also want to take a look at this Bourbaki exposition by Bloom on the work of Manners on a new proof of GI(s) with better numerical implications. https://arxiv.org/abs/2009.01774

### Part 6: Back to linear equations

Follow the path from GI(s) and MN(s) to the main result more closely than done in Part 1.

# Seminar program - season 2

We are following the Tao-Teräväinen paper: https://arxiv.org/abs/2107.02158 .

Tao's blog post about the above paper might also help: https://terrytao.wordpress.com/2021/07/05/

Part III

### Motivation

Write $Q:= \exp ( (\log N)^{1/10} )$.

Note that Q defined in this way grows slower than any positive power of $N$ because $N^{c} = \exp (c \log N)$ and $c\log N > (\log N )^{1/10}$. It grows faster than any power of $\log N$ because $\log ^A N := (\log N)^A = \exp (\log ( \log ^A N ) ) = \exp ( (\log \log N ) + \log A )$ and we know $(\log N )^{1/10 } > \log \log N$.

The main result is the bound of the following Gowers uniformity norm, with some sufficiently small $c>0$:

$\| \Lambda - \Lambda _{Cramér,Q} \| _{U^k ([N])} \ll (\log \log N )^{-c} +Q^{-c} .$

Let's write $\| - \| := \| - \| _{U^k([N])}$ to make it easier to type.

By the triangle inequality of $\| - \| _{U^k}$, this would follow from:

$\| \Lambda - \Lambda _{Siegel} \| \ll (\log \log N)^{-c} \quad \text{and }\quad \| \Lambda _{Siegel} - \Lambda _{Cramér,Q} \| \ll Q^{-c} .$

The latter inequality is easier and proved in §5.

The former requires substantial work.

By Manners' form of GI(s) explained in §6, this will follow from Theorem 2.7:

for certain nilsequences $\mathbb Z \xrightarrow{g} G\to G/\Gamma \xrightarrow{F}\mathbb C$ and arithmetic progressions $P\subset [N]$, we have

$\sum _{n\in P} (\Lambda - \Lambda _{Siegel})(n) \overline F (g(n) ) \ll N\exp ( (-\log N)^{0.09 } ) .$

This estimate is the main part of the paper. The proof requires §5 + §7 + former Green-Tao work on MN(s).

By the way, §1 of the paper is introduction, §2 is an extended introduction including def of $\Lambda _{Siegel}$, §3 is a one-page section about the notation, §4 is recollection of $\| - \| _{U^k([N])}$.

The main content spans through §§5-8 (as we will see below). §9 explains the application to linear equation in primes. The final section §10 explains another application.

### Seminar Program - Part I

The following topics can be studied independently for the time being, so let's distribute them and give lectures to each other:

#### Basics of $\Lambda _{Cramér,w}$

Cover §5.1. The computation is annoying. But eventually we'll have to be comfortable with this type of computation.

#### Basics of $L (s,\chi )$ and Siegel zeros

What is $L(s,\chi )$, to begin with? (Read §5.9 of the Iwaniec-Kowalski book, say.)

What are the (Landau-)Siegel zeros? (Read p.106 of the Iwaniec-Kowalski book, say.)

Theorem 2.7 (p.122) of that book (which should motivate $\Lambda _{Siegel}$).

Wikipedia would help too. https://en.wikipedia.org/wiki/Siegel_zero

#### Manners' GI(s)

Recall relevant notions and state Theorem 6.2 of Tao-Teräväinen. (I believe all the relevant notions can be found within the paper.)

[optional] Explain the implication (Theorem 2.7 of Tao-Teräväinen) $\Rightarrow$ $\| \Lambda - \Lambda _{Siegel} \|_{U^k([N])} \ll (\log \log N )^{-c}$, perhaps ignoring the "boundedness" issue for now.

[optional but preferred] What would be Manners' GI(s) for $\mathbb Z^n$ if we implement this blog post of Tao? https://terrytao.wordpress.com/2015/07/24/

### Part II

Having gone through Part I, we will be ready to study the following topics. Again, they can be studied independently of each other.

#### Toward the conparison of $\Lambda _{Siegel}$ and $\Lambda _{Cramér,Q}$

Present the first half of §5.2 (p. 20 -- first quarter of p. 22).

This seems to depend on what we have learned about Siegel zeros in Part I.

#### The first 1/3 of §7

Here, they prove Prop. 2.2, the main theorem with the test function $F$ constant.

We want to understand this part in depth.

#### Manners' GI(s) continued

Read §8, where they modify Manners' GI(s) so that it can be applied to non-bounded functions like $\Lambda - \Lambda _{Siegel}$.