new plans

[Liu]

I realize I can’t control the progress of each learning process. So I
decided to start my two new sub plans at the same time and I will not
give them time restrictions. I gether all my sub plans at a post of my
blog and stick it in the front page. http://liuxiaochuan.wordpress.com/2009/12/13/math-learning-plans/

Sub plan on Szemeredi’s theorem:

I will start learning Szemeredi’s theorem systematically, which will
be a very long process. Szemeredi’s theorem can be regarded as one of
the biggest theorem in Additive Combinatorics. I will try to
understand it from different aspects. A good place to start will be
the book:”Additive Combinatotics” written by Drs. Terence Tao and Van
Vu. I think reading this theorem will at least take several weeks or
even months, so this plan is pretty big in some sense. It will depend
on several other plans.

Sub plan on Finite Fourier Analysis:

Dirichlet’s theorem states that if q and l are to coprime positive
integers, then the following sequence must contain infinite primes:

$latex l,l+q,l+2q,\cdots$

The proof of this theorem is based on a finite fourier analysis. The
last two chapters of the book:"Fourier Analysis" by E.Stein and
R.Shakarchi are dealing with this problem.

[Akhil Mathew]

Liu,

Unfortunately I haven't read Stein's book (and Google Books and amazon have
no preview) but I'm pretty sure the core of the usual proof of Dirichlet's
theorem is showing that the L-functions associated to non-trivial characters
don't vanish at s=1. Only a couple of facts about Dirichlet characters are
needed. The analytic continuation of the L-functions to Re s>0 is basically
summation by parts.

Serre's _A Course in Arithmetic_ has a really clean, crisp exposition of
this.

Incidentally, the stronger version of Dirichlet's theorem (that gives the
regular density of primes in an arithmetic progression) can be proved
similarly as the usual prime number theorem; see

http://www.math.umass.edu/~isoprou/pdf/primes.pdf

for a short exposition based on the article by Newman on the prime number
theorem.

Akhil




----- Original Message -----
From: "Liu" <lxc...@gmail.com>
To: "maths learning" <maths-l...@googlegroups.com>
Sent: Sunday, December 13, 2009 7:00 AM
Subject: new plans


I realize I can�t control the progress of each learning process. So I

decided to start my two new sub plans at the same time and I will not
give them time restrictions. I gether all my sub plans at a post of my
blog and stick it in the front page.
http://liuxiaochuan.wordpress.com/2009/12/13/math-learning-plans/

Sub plan on Szemeredi�s theorem:

I will start learning Szemeredi�s theorem systematically, which will
be a very long process. Szemeredi�s theorem can be regarded as one of

the biggest theorem in Additive Combinatorics. I will try to
understand it from different aspects. A good place to start will be

the book:�Additive Combinatotics� written by Drs. Terence Tao and Van

Vu. I think reading this theorem will at least take several weeks or
even months, so this plan is pretty big in some sense. It will depend
on several other plans.

Sub plan on Finite Fourier Analysis:

Dirichlet�s theorem states that if q and l are to coprime positive

[Liu]

Akhil,
yeah, you are good.I almost finish reading the proof. In order to
prove the non-vanishing of L-function for non trivial characters,
Stein's book uses a very elementary method, dividing into two
situations where charaters are all taking real values or not.

I also want to cover some general Fourier-analystic methods in
Additive Combinatorics later.

On Dec 13, 11:18 pm, "Akhil Mathew" <akhilmat...@verizon.net> wrote:
> Liu,
>
> Unfortunately I haven't read Stein's book (and Google Books and amazon have
> no preview) but I'm pretty sure the core of the usual proof of Dirichlet's
> theorem is showing that the L-functions associated to non-trivial characters
> don't vanish at s=1.  Only a couple of facts about Dirichlet characters are
> needed.  The analytic continuation of the L-functions to Re s>0 is basically
> summation by parts.
>
> Serre's _A Course in Arithmetic_ has a really clean, crisp exposition of
> this.
>
> Incidentally, the stronger version of Dirichlet's theorem (that gives the
> regular density of primes in an arithmetic progression) can be proved
> similarly as the usual prime number theorem; see
>
> http://www.math.umass.edu/~isoprou/pdf/primes.pdf
>
> for a short exposition based on the article by Newman on the prime number
> theorem.
>
> Akhil
>
> ----- Original Message -----
> From: "Liu" <lxc1...@gmail.com>
> To: "maths learning" <maths-l...@googlegroups.com>
> Sent: Sunday, December 13, 2009 7:00 AM
> Subject: new plans
>

> I realize I can t control the progress of each learning process. So I

> decided to start my two new sub plans at the same time and I will not
> give them time restrictions. I gether all my sub plans at a post of my

> blog and stick it in the front page.http://liuxiaochuan.wordpress.com/2009/12/13/math-learning-plans/
>
> Sub plan on Szemeredi s theorem:
>
> I will start learning Szemeredi s theorem systematically, which will
> be a very long process. Szemeredi s theorem can be regarded as one of

> the biggest theorem in Additive Combinatorics. I will try to
> understand it from different aspects. A good place to start will be

> the book: Additive Combinatotics written by Drs. Terence Tao and Van

> Vu. I think reading this theorem will at least take several weeks or
> even months, so this plan is pretty big in some sense. It will depend
> on several other plans.
>
> Sub plan on Finite Fourier Analysis:
>

> Dirichlet s theorem states that if q and l are to coprime positive