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- Mar 22, 2013

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

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- Mar 22, 2013

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

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- Mar 22, 2013

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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 wont repeat and get frowned at )

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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- Jan 17, 2013

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- Mar 22, 2013

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

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?

- Jan 17, 2013

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Of course ! all these seem interesting topics !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?

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- Mar 22, 2013

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

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.

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- Mar 22, 2013

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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?

- Jan 17, 2013

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

Right ?

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- Mar 22, 2013

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I am not sure exactly what proof you are asking for. Could you elaborate?ZaidAlyafey said:It would be so nice if you can sketch the proof of the eta representation using the zeta

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- Jan 17, 2013

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

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

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- Mar 22, 2013

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

Yes, I used the reflection formula of gamma and zeta to get a functional equation of H-L eta.ZaidAlyafey said:I think you are some how using the reflection formula of the Gamma function

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- Mar 22, 2013

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(*) : 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).

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- Mar 22, 2013

- 573

I am leaving this notification here for an announcement.

Balarka

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