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I see that it is proven by Hardy in 1914 that there are an infinite number of zeros on the critical line. I also see that the Hardy and Littlewood conjectures appear in some papers they wrote together in 1921. However, it was further proven by Littlewood alone that if $\gamma_n$ is an increasing sequence of the imaginary parts of the zeros on the critical line in the upper complex half-plane then

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

Can someone please give me the citation for the paper in which Littlewood proved this result on his own? I absolutely cannot find it!

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    $\begingroup$ Look in Titchmarsh's book where a proof is given, and you'll also find a reference to Littlewood's paper. $\endgroup$
    – Lucia
    Commented Nov 11, 2019 at 17:30
  • $\begingroup$ Thanks! Could you send me a link to the PDF of that book with the page number? $\endgroup$
    – Jimmy Rankerman
    Commented Nov 11, 2019 at 17:37
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    $\begingroup$ Two notes on the Riemann zeta-function, Proc. Camb. Phil. Soc. 22 (1924), 234-242. doi.org/10.1017/S0305004100014158 $\endgroup$
    – MyNinthAccount
    Commented Nov 11, 2019 at 18:16

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