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# Mitsui's prime number theorem

I am being forced to read Mitsui's paper. https://www.jstage.jst.go.jp/article/jjm1924/26/0/26_0_1/_pdf 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…

# 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/ Table of Contents Motivation Seminar Program - Part I; I-1, I-2…

# Ben Green's lectures on nilsequences

Go back to the program https://motivichomotopy.hatenablog.jp/entry/2020/10/19/171757 Green's page about the videos http://people.maths.ox.ac.uk/greenbj/videos.html Lecture 1 http://www.youtube.com/watch?v=HM3jR0b4VHY 1:18:00 Explains the v…

# The norm of polynomial maps

This excerpt from https://arxiv.org/abs/1311.6170 summarizes basic facts on the norm of polynomial maps. Recall that the norm $\Vert \theta \Vert _{R/Z}\in [0,1/2]$ of a real number $\theta$ is the distance from $\theta$ to the closest i…

# メビウス論文 Type I case の証明の検討

https://dx.doi.org/10.4007/annals.2012.175.2.3 のp.549を読んでいます。 個数$\gg \delta ^{O(c_1)} K$個以上の$k\in (K,2K]$に対して不等式 $\mathbb E _{w, N^{0.9}<kw\le N } 1_{P}(kw) F(g(kw) ) \gg \delta ^{O(c_1)}$ が成り立つ状況を考えています。$P\subset [N]$は等差数列です。$c_1>0$は定理2.1の$c>0$に対応する、あとで決める小さな正の数です。 初めから$kw\in P$</kw\le>…

# Divisor function

Let $K$ be a number field. For a non-zero ideal $\mathfrak a$, write $t(\mathfrak a)$ for the number of its divisors (in the multiplicative monoid of non-zero ideals). It's called the divisor function. I wanted to type tau but as always ha…

# Vaughan's decomposition 2

In the previous post we estimated the first term on the right hand side of: $\mathbb E_{N^{0.9} < n \le N }\mu (n)\bar f(n) = - \mathbb E_{N^{0.9}<n\le N } \sum _{b\le U,c\le V , bc|n } \mu (b) \mu (c) \bar f(n) + \mathbb E_{N^{0.9}<n\le N } \sum _{b> U,c> V , bc|n } \mu (b) \mu (c) \bar f(n) .$…</n\le>

# Vaughan's decomposition

In this post I will perform a routine task of verifying something routine. Let $U,V,N$ be positive integers satisfying $UV < N^{0.9}$. The exponent here can be replaced by any positive number $<1$ but I don't want to complicate the notatio…

# Green-Tao-Ziegler theorem implies that for the localized integers

In a paper on rational points on varieties, they exploit the Green-Tao-Ziegler theorem. One way to state the theorem is as follows. Let $L_i(x,y)\in \mathbb Z [x,y]$ (i=1, ..., r) be finitely many homogeneous polynomials of degree 1 and as…

# 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…

# Gowersノルムと、逆予想の逆

Gowersノルムは、整数$s\ge 0$を固定するごとに与えられます。有限アーベル群$Z$上の関数$f\colon Z\to \mathbb C$に対して定義されます。$\Vert f \Vert _{U^{s+1} (Z) }$は次の値の$2^{s+1}$乗根です：\[ \Vert f \Vert _{U^{s+1}(Z)} ^{2^{s+1} }:= \math…

# 単連結な冪零リー群についての事実まとめ

Baker-Campbell-Hausdorffの定理というものがあります (Wikipedia)。$G$をリー群、$\mathfrak g$をそのリー環、$X\mapsto e^X$をその指数関数とするとき、$X,Y\in \mathfrak g$が十分原点に近い範囲で、\[ e^X e^Y =e^Z \text{ と書く時, }Z= X+Y+\frac 1 2 …