Timeline for The Hardy Z-function and failure of the Riemann hypothesis
Current License: CC BY-SA 2.5
10 events
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| Apr 10, 2011 at 10:28 | comment | added | Marc Palm | The Riemann Zeta function is generically unbounded and known to be quasiperiodic if the Riemann hypothesis holds. Even independent of RH approximates itsself by positive lower density everywhere where the Riemann zeta function does not vanish. But i think the question is more natural to ask for $\zeta$ first and then answer them for $Z$. | |
| Dec 27, 2010 at 4:59 | vote | accept | David Hansen | ||
| Dec 24, 2010 at 21:46 | comment | added | David Hansen | Terry: Thanks very much for that clarification! I think I understand now why my conjecture might be considered "unnatural" - which is not to say I don't believe it's true! :) | |
| Dec 23, 2010 at 19:29 | comment | added | Terry Tao | It is technically possible to be both locally quasiperiodic and generically unbounded; but generic unboundedness makes the local quasiperiodicity property a much less natural property to ask for - it's like asking for recurrence in a dynamical system where the orbits are going to infinity on the average, as opposed to mostly being confined to a compact set. Recurrence is of course still possible in such situations (e.g. for Brownian motion in one and two dimensions) but becomes a much "thinner" property: one would no longer expect to obtain recurrence on a set of times of positive density. | |
| Dec 23, 2010 at 18:14 | comment | added | David Hansen | Hmm, yeah, that was a poor interpretation. I guess what I meant was that $Z(t)$ is unbounded very "mildly", perhaps so mildly that local quasiperiodicity is still plausible. Must a locally quasiperiodic function be bounded on average? That doesn't seem obvious. | |
| Dec 23, 2010 at 18:12 | history | edited | Terry Tao | CC BY-SA 2.5 |
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| Dec 23, 2010 at 18:05 | comment | added | Terry Tao | I'm not sure what you mean by ""almost always" bounded" here; Soundararajan's corollary B has a logarithmic loss in it. Selberg's law tells us that the typical value of Z(t) is like $\exp( c \sqrt{\frac{1}{2} \log \log t} )$, where c fluctuates normally. This does not seem particularly consistent with local quasiperiodicity as you have defined it. | |
| Dec 23, 2010 at 17:57 | comment | added | David Hansen | Thanks for the comments, Terry! I don't think you need to appeal to Selberg for the unboundedness - Hardy-Littlewood showed $\int_{0}^{T}|\zeta(1/2+it)|^2 dt \sim T\log{T}$ about twenty-five years before Selberg's work. Even granted this, $Z(t)$ is "almost always" bounded (see for example Corollary B in Soundararajan's paper "Moments of the Riemann zeta function"), so I'm not sure if a rescaling is necessary... | |
| Dec 23, 2010 at 17:53 | history | edited | Terry Tao | CC BY-SA 2.5 |
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| Dec 23, 2010 at 17:43 | history | answered | Terry Tao | CC BY-SA 2.5 |