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I recently asked for the original journal citation on Littlewood's result

$$ \lim\limits_{n\to\infty}|\gamma_n-\gamma_{n-1}| =0 ~~,$$

wherein $\gamma_n$ is an increasing sequence of the imaginary parts of the zeros the RZF in the upper complex half plane. Thank you very much to this great community for providing the original paper, but it has proven a little dense for me to work through. I understand that this result is also proven in Titchmarsh (1986), but I was not able to find it in there. If someone would be so kind as as to direct me toward a resource which meticulously develops Littlewood's above result (or if you have the page number in Titchmarsh), preferably at the level of an advanced undergraduate, then I would be very happy to see it. Thank you very much!

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The result you want is in article 9.12 of Titchmarsh. This is on page 191 of the first edition and page 224 of the new Heath_Brown edition. This result is unconditional, whereas Littlewood's best result assumes R.H. so is stronger.

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  • $\begingroup$ Thanks a lot for this. I am also looking for the 1914 theorem of Hardy that shows $\zeta(1/2+it)$ has infinitely many real zeros. Do you know where this is in Titchmarch 1986? It is so dense, I am really having trouble perusing it! Thank you so much!!! $\endgroup$
    – Jimmy Rankerman
    Commented Nov 27, 2019 at 18:38
  • $\begingroup$ it's article 10.2 on p. 256 $\endgroup$
    – TPTW
    Commented Nov 28, 2019 at 21:42

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