Timeline for Average gap between zeros on the critical strip of the Riemann Zeta Function
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 6, 2021 at 21:08 | comment | added | David Farmer | @WillSawin Yes, thanks. I should have said "...*if* the de Bruijn–Newman constant is negative." and later "...proved the de Bruijn–Newman constant is non-negative." | |
| Feb 5, 2021 at 0:33 | comment | added | Will Sawin | @DavidFarmer Excellent summary of the situation! A small correction - Tao and Rodgers only proved the constant is greater than or equal to 0, not exactly 0. (As it would have to be, given that their argument is unconditional of RH, because the constant being less than or equal to 0 is equivalent to RH). | |
| Feb 3, 2021 at 13:44 | comment | added | David Farmer | 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. | |
| Feb 3, 2021 at 4:26 | comment | added | Gerry Myerson | 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. | |
| Feb 3, 2021 at 3:16 | comment | added | Gerry Myerson | It seems unlikely that the zeta zero locations behave like an arithmetic progression. Can you provide a link to the Tao lecture? | |
| Feb 2, 2021 at 23:43 | comment | added | Trevor Krumrine | I saw terence tao's lecture on youtube about the distribution of the gaps. He referred to these gaps as being either water, solid, or gas to specify that the gaps are either not equally space on the real line = water, on the real line and equally spaced = solid, or lie off the real line = gas. He also showed that the Riemann zeta function behaves like Bessel functions asymptotics for positive t in other words the zeros of the function behave like arithmetic progressions after passing a certain threshold e(c / t) where c is constant.He also found a more generalized Riemann--Siegel approximation | |
| Feb 2, 2021 at 22:38 | comment | added | Stopple | 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. | |
| Feb 2, 2021 at 22:29 | comment | added | მამუკა ჯიბლაძე | 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) | |
| Feb 2, 2021 at 22:21 | comment | added | მამუკა ჯიბლაძე | (For more detail and references see en.wikipedia.org/wiki/Riemann_hypothesis#Number_of_zeros) | |
| Feb 2, 2021 at 21:47 | comment | added | Gerry Myerson | The number of zeroes up to height $T$ is, asymptotically, ${T\over2\pi}\log{T\over2\pi}$. | |
| Feb 2, 2021 at 21:26 | history | asked | Trevor Krumrine | CC BY-SA 4.0 |