Are there any approximations of the average gap between 2 successive zeros along the Riemann zeta functions critical strip up to the nth zero? If so, is it hypothesized that this average gap converges and if it doesn't then what does this mean for the primes?
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5$\begingroup$ The number of zeroes up to height $T$ is, asymptotically, ${T\over2\pi}\log{T\over2\pi}$. $\endgroup$– Gerry MyersonCommented Feb 2, 2021 at 21:47
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1$\begingroup$ Also note that $(\gamma_2-\gamma_1)+(\gamma_3-\gamma_2)+...+(\gamma_n-\gamma_{n-1})=\gamma_n-\gamma_1$, so essentially you are just asking about the limit of $\gamma_n/n$ (which is zero, as follows from the above formula) $\endgroup$– მამუკა ჯიბლაძეCommented Feb 2, 2021 at 22:29
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1$\begingroup$ There is a lot of interest in the distribution of the gaps, see (for just one example) arxiv.org/pdf/0909.4914.pdf for an overview. $\endgroup$– StoppleCommented Feb 2, 2021 at 22:38
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1$\begingroup$ OK, I think you're referring to youtu.be/t908N5gUZA0 What he says is that if you look at a small interval far out, e.g., between $T$ and $T+1$ for $T$ much larger than $1$, then the zeros in that interval will be evenly spaced – not exactly evenly, but close to evenly. But I think even that conclusion is conditional on the Riemann Hypothesis. $\endgroup$– Gerry MyersonCommented Feb 3, 2021 at 4:26
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1$\begingroup$ In the cited lecture Terry Tao does not say that the zeros of the zeta function will be close to evenly spaced, nor did he say that the zeros behave like an arithmetic progression. He said that the zeros behave in that (absurd) way if the de Bruijn–Newman is non-zero. But the zeros are known to not behave in that absurd way, so this is a contradiction. That is how Tao and Rodgers proved the de Bruijn–Newman constant equals zero. The proof does not assume the Riemann Hypothesis. $\endgroup$– David FarmerCommented Feb 3, 2021 at 13:44
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