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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 


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


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


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


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


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."




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
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 
and 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

- Make some comments on proofs of some of them, or discuss examples, or draw pictures. 



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.


how the statement is reduced to the equi-distributed case



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.


Mitsui's prime number theorem

I am being forced to read Mitsui's paper.


Mitsui's prime number theorem

The main theorem of this paper says, in a simpified form, the following.

To state it, let $K$ be a number field and $\Vert - \Vert \colon O_K \to \mathbf R _{\ge 0}$ be the norm $\alpha \mapsto \max\limits _{ \iota \colon K\hookrightarrow \mathbf C} |\iota  (\alpha ) | $.

Theorem (Main Theorem on p. 35)

Let $\mathfrak a\subset O_K$ be a non-zero ideal and $b\in O_K$ an element coprime to $\mathfrak a$. For positive numbers $X$, denote by $\pi (\mathfrak a ,b ,X)$ the number of prime elements $\alpha \in O_K$ which are congruent to $b$ modulo $\mathfrak a$ such that $\Vert \alpha \Vert \le X $.

Then as $X\to +\infty $ we have 

\[ \pi (\mathfrak a ,b , X) = \frac{1 }{\varphi (\mathfrak a )\sqrt{|D_K|}\cdot  \mathrm{res} _{s=1} \zeta _K(s) }  \int _2^X \cdots \int _2^{X^2} \frac{\mathrm{d}t_1\cdots \mathrm dt_{r_1+r_2} }{\log (t_1\cdots t_{r_1+r_2} ) }  + O(X^n e^{-c \sqrt{n \log X} } ) ,  \] 

where we integrate $t_i$ up to $X$ if $1\le i\le r_1$ (the number of real embeddings) and up to $X^2$ if $r_1<i\le r_1+r_2$, and the constant $c>0$ depends only on the field $K$.

I suppose the value of the integral in the above formula is roughly equal to $X^n / (n\log X)$.


Structure of Mitsui's paper

Besides the unnumbered introduction, the paper consists of 4 sections.

In Section 1 (pp. 2-13), he proves the following asymptotic formula for the number of prime ideals of bounded norms (wasn't it known?): if $C \in Cl (K,\widetilde {\mathfrak a} )$ is a class in the ray class group with modulus $\widetilde{\mathfrak a} $, then as $x\to +\infty $, we have the estimate of the number of prime ideals belonging to $C$ in $Cl (K,\widetilde{\mathfrak a} ) $ and having norms $\le x$ :

\[ \pi (x,C )=\frac{1}{ | Cl (K, \widetilde{\mathfrak a} ) | } \int _2^x \frac{\mathrm d t}{\log t} + O(x e^{-c \sqrt{\log x} } ) .   \] 

In Section 2 (pp. 13-20), titled "Grössencharacter and preliminary lemmas," he spends 7 pages to do whatever preliminary that he needs later.

Section 3 spans through pp. 20-35 and is the technical heart. The main result of this section (p. 20) allows one to count the prime numbers in cone-shaped regions.

Section 4 consists of pp. 35-42. Here he deduces the main theorem above. He splits the cube $\{ \alpha \mid \Vert \alpha \Vert \le X \} $ into small quadrangular pyramids and applies the result of Section 3 to each of these pyramids.

I will discuss some details of Sections 3 and 4 below.


Section 3

I have to set up some notation to discuss the results of this section.

Recall that $r_1$ is the number of real embeddings $K\hookrightarrow \mathbf R $  and $r_2$ is the number of conjugate pairs of complex embeddings $K\hookrightarrow \mathbf C $. For the notational convenience, for an embedding $\iota \colon K\to \mathbf C $ let $| - | _\iota $  be the absolute value of $\iota (\alpha ) $ if $\iota $ is a real embedding, and $|\alpha |_\iota := |\iota (\alpha ) |^2$ if $\iota $ is complex.

Consider the multiplicative Minkowski map 

\[ O_K \setminus \{ 0\} \to \mathbf R ^{r_1+r_2} \]

defined by $\alpha \mapsto (\log | \alpha  |_\iota  )_{\iota \colon K\to \mathbf C}$.


Dirichlet's theorem says that the image of $O_K ^*$ is a lattice in the hyperplane $H= \{ (x_1,\dots ,x_{r_1+r_2}  ) \mid x_1+\dots +x_{r_1+r_2}=0 \} \subset \mathbf R^{r_1+r_2} $.

Then Theorem on p. 20 and Corollary on p. 35 says the following:

The number of prime elements having norms $\le x$ and lying in the depicted region is proportional to the area of the chosen region on the hyperplane $H$.


(If the chosen region is a fundamental paralellopiped, then the estimate is euqual to that in Theorem in Section 1.)


I guess now my task is to pin down exactly how and where analysis turns into the estimate of the number of prime elements. See you!



Section 4

Let me explain how he splits up the region $\Vert - \Vert \le X$.