Hardy & Littlewood's Estimate Result

[mathbalarka]

Introduction :

This thread is dedicated to discuss about Hardy-Littlewood's estimate of the \(\displaystyle N_0(T)\), i.e., the number of critical zeros of the Riemann zeta function with imaginary part smaller than \(\displaystyle T + 1\). The final result of Hardy-Littlewood estimate shows that there are infinitely many zeros of zeta that lies in the critical line.

This thread would be continued in more than one post, each of them showing different lemmas. The estimated number of post required would be 8 or less. There could be prolonged gaps in between the posts which are to be continued, so it would certainly be disappointing not to be able to look at it all at once; but I think a serious subject like this could take days to digest, so go slow and you won't throw up! :D Nevertheless, I think this note would be of particular interest once completed.

And as a final note, I would be putting it up altogether in a very short, less elegant and more understandable format. I have put quite a few things from here and there, so there could possibly be many typos. I hope members of MHB would certainly help me to point those out.

The commentary thread for this note is here : http://mathhelpboards.com/commentary-threads-53/commentary-hardy-littlewoods-result-7375.html

Balarka
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[mathbalarka]

The topic is for discussion of the tutorial :

http://mathhelpboards.com/math-notes-49/hardy-littlewoods-result-7374.html

 

[mathbalarka]

I had been thinking about posting this tutorial for a while. I often hear from peoples that they don't even get what RH is basically saying, let alone the fact that they understand this major result in the theory of L-functions (an extension of this to L-forms exists, I think).

So I decided I would master myself in the theory of zeta functions first and then come back here to post the tutorial. Although there are a whole lot of things I don't understand even after a year of study, I think I am now quite capable of posting a good and explicit tutorial here.

Anyways, I don't think I would be able to get started too soon, for obvious reasons (many peoples at this site know my so called "obvious reasons" so I won't repeat and get frowned at :p)

Well, the upper bound on the date of the next post is 20th Nov, however, I believe I would get plenty of times before that, I am quite sure.

PS I know many peoples would be wondering then why I didn't start in in 20th Nov if I don't have time right now. The answer from me would be to keep our readers in a little suspense ;)

Balarka
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[alyafey22]

I will be interesting to see your tutorial ( disappointed it will take that long ) .Anyways , I am sure it will be so interesting for me as long as you don't include so much number theory stuff (Dull) . It would be interesting if you post an outline of the topics you are going to discuss and the background needed for such tutorial .

 

[mathbalarka]

It would be interesting if you post an outline of the topics you are going to discuss and the background needed for such tutorial .

Actually, if you peer at Titchmarsh & Health-Brown, you'll see several complicated proof by ACing etas and applying Real and Complex analysis to it. The estimates are also gigantic. But the proof I am posting must be a refinement by Iwaniec & Kowalski; it's pretty neat, you know.

The basic things you will need is a very good understanding of the analytic continuation of zeta on the critical strip. Furthermore, you need to understand the functional equation of zeta, some basic estimates of zeta on the critical line. Along with these, you need to understand the basics of asymptotic analysis and integral analysis (the whole proof relies mostly on integral analysis and estimates).

PS After this tutorial is complete, I'd think about posting another tutorial which will consist of the Selberg's result (a positive proportion of the zeros of zeta lies in the 1/2-line) and merge those tutorials. Would it be a good idea?

 

[alyafey22]

PS After this tutorial is complete, I'd think about posting another tutorial which will consist of the Selberg's result (a positive proportion of the zeros of zeta lies in the 1/2-line) and merge those tutorials. Would it be a good idea?

Of course ! all these seem interesting topics !

 

[mathbalarka]

Now, about the approach, we would be considering the real-valued eta function instead of the zeta, i.e.

\(\displaystyle \eta(s)= \frac{\pi^{-1/4 - is/2} \, \Gamma(1/4 + is/2)}{\left | \pi^{-1/4 - is/2} \, \Gamma(1/4 + is/2) \right |} \zeta\left (\frac{1}{2} + i u \right )\)

Next, we consider the integrals \(\displaystyle I_0(z) = \int_{t}^{t + \Delta} \eta(s) \, \mathrm{d}s\) and \(\displaystyle I_1(z) = \int_{t}^{t + \Delta} \left | \eta(s) \right | \, \mathrm{d}s\) for some positive \(\displaystyle \Delta\)

The basic idea of the proof is to show that \(\displaystyle \left | I_0(t) \right | < I_1(t)\) infinitely often regardless of \(\displaystyle \Delta\); in fact, this is equivalent to show that there are infinitely many zeros of zeta which lies in in the 1/2-line.

 

[mathbalarka]

Introducing The Lemmas (1) :

First, we introduce a transform of zeta at the half line, i.e., the eta function -

\(\displaystyle \eta(s)= \frac{\pi^{-1/4 - is/2} \, \Gamma(1/4 + is/2)}{\left | \pi^{-1/4 - is/2} \, \Gamma(1/4 + is/2) \right |} \zeta\left (\frac{1}{2} + i u \right )\)

Note that the Hardy-Littlewood eta has a very neat functional equation \(\displaystyle \pi^{-s/2} \Gamma(s/2) \eta(s) = \pi^{-(1-s)/2} \Gamma((1-s)/2) \eta(1-s)\) which implies that eta is also even.

The first integral we are interested in is \(\displaystyle I_1(t) = \int_{t}^{t + \Delta} \left | \eta(s) \right | \, \mathrm{d}s\). We preasent a non-trivial lemma regarding this eta-integral :

Proposition : There exists a function \(\displaystyle K(t, \Delta)\) such that \(\displaystyle I_1(t) \ge \Delta - K(t, \Delta)\) and \(\displaystyle \int_{T}^{2T} \left | K(t, \Delta) \right |^2 \,\, \mathrm{d}t \ll T\) where \(\displaystyle T \ge \Delta^2 \ge 1\)

The first portion can be proved very easily, by elementary integral analysis :

\(\displaystyle \begin{aligned}I_1(t) = \int_{t}^{t + \Delta} \left | \eta(s) \right | \, \mathrm{d}s &= \int_{0}^{\Delta} \left | \zeta(1/2 + it + iu) \right | \, \mathrm{d}u\\ &\ge \left | \int_{0}^{\Delta} \zeta(1/2 + it + iu) \, \mathrm{d}u \right |\\ &\ge \Delta - \left | \int_{0}^{\Delta} (\zeta(1/2 + it + iu) - 1) \, \mathrm{d}u \right |\\ &= \Delta - K(t, \Delta) \end{aligned}\)

Thus, proving the first part. We now proceed to prove the second part, which is considerably harder than the former -

We choose the formula \(\displaystyle \zeta(s) = \sum_{n \leq T} \frac{1}{n^s} + \mathcal{O} \left (T^{-1/2} \right )\) for \(\displaystyle \Re = 1/2\) to estimate \(\displaystyle K(t, \Delta)\) as

\(\displaystyle K(t, \Delta) = \left | \sum_{n \leq T} n^{\frac{-1}{2} - it} \frac{1 - n^{-i \Delta}}{\log n} \right | + O(\Delta T^{-1/2})\)

Now we use a well-known theorem on integral mean-value estimates over Dirichlet polynomials. The proof can be easily worked out by using tools of integral analysis and asymptotic analysis, thus we do not omit it here :

Theorem : For any complex sequence \(\displaystyle a_n\), \(\displaystyle \int_{0}^{T} \left | \sum_{n \leq N} a_n n^{i t} \right |^2 \,\, \mathrm{d}t = (T + \mathcal{O}(N)) \sum_{n \leq N} | a_n |^2\)

By a little tweaking of the above formula, we get the desired result. In fact, we even have the asymptotic constant \(\displaystyle \sum_{n \leq T} \frac{1}{n (\log n)^2}\).

 

[mathbalarka]

Hopefully, I had time before Nov 20 and posted a lemma already.

I have omitted a proof of a certain theorem related to the result. I was wondering whether to give it or not. What is the reader's opinion on it?

 

[alyafey22]

It would be so nice if you can sketch the proof of the eta representation using the zeta function also I think you have a typo in the functional equation it should be

\(\displaystyle \displaystyle \pi^{-s/2} \Gamma(s/2) \eta(s) = \pi^{-(1-s)/2} \Gamma((1-s)/2) \eta(1-s)\)

Right ?

 

[mathbalarka]

It would be so nice if you can sketch the proof of the eta representation using the zeta

I am not sure exactly what proof you are asking for. Could you elaborate?

 

[alyafey22]

I meant the relation

\(\displaystyle \displaystyle \eta(s)= \frac{\pi^{-1/4 - is/2} \, \Gamma(1/4 + is/2)}{\left | \pi^{-1/4 - is/2} \, \Gamma(1/4 + is/2) \right |} \zeta\left (\frac{1}{2} + i u \right )\)

I think you are some how using the reflection formula of the Gamma function and \(\displaystyle f\cdot \overline{f}=|f|^2\)

 

[mathbalarka]

I meant the relation ...

It's not a relation, it's a definition. Hardy & Littlewood defined a function (Hardy Littlewood eta) which is related to zeta like that.

I think you are some how using the reflection formula of the Gamma function

Yes, I used the reflection formula of gamma and zeta to get a functional equation of H-L eta.

 

[mathbalarka]

The second result that'd be shown in the tutorial thread would require the use of convexity bounds for zeta. I'll post the proof of that particular non-trivial bound later as a note and also give a proof of subconvexity bound as it could be useful for Selberg's result, as well as Conrey's approach (*)

(*) : The theorem shows what Selberg's doesn't -- the precise proportion of zeros in the critical line. It can be shown that there are 40% of the zeros of zeta which lies on the 1/2-line. Note that to show Conrey's approach, one must go through Levinson's (33.3% of the zeros of zeta lies on the 1/2-line).

 

[mathbalarka]

I was thinking about posting it since 19-th Nov, but because of unfortunate events, the second one post is postponed till 24-th. I have an appointment with a professor at 24-th, so you know, preparing a bit by revising my skills (not confident).

I am leaving this notification here for an announcement.

Balarka
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