Economics Job Market Rumors Topic: A proof of the Riemann Hypothesis on the horizon.
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon
Economics Job Market Rumors Topic: A proof of the Riemann Hypothesis on the horizon.en-USSat, 01 Apr 2023 10:16:45 +0000http://bbpress.org/?v=1.0.2<![CDATA[Search]]>q
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Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8629457
Fri, 06 Jan 2023 22:02:51 +0000Economist8629457@https://www.econjobrumors.com/<p>Alessandro is a visiting postdoc working with Michael Tschantz in the Usable Security and Privacy group. He earned a MASt from the University of Cambridge, UK, and a Ph.D. from Stanford. His math research is in geometry and number theory and he has been progressively becoming interested in applied aspects of mathematics. His current research while at the Institute is in applications of game theory to differential privacy.
</p>Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8505785
Wed, 23 Nov 2022 12:51:32 +0000Economist8505785@https://www.econjobrumors.com/<blockquote><p>So are we settled that this is a gap?
</p></blockquote>
<p>No. The main mistake you're making is that, it seems you'rethinking that the expression in (18) is real-valued, but it's actually not. That is, it is not a C \mapsto R function, but it is complex-valued and indeed analytic for sigma in C, Re(sigma) > Theta.</p>
<p>It seems you're being mislead by the Re(...) and Im(...) in the brackets to thinking that the expression is real-valued. However, the correct way to see it is that, </p>
<p>Re(x^{-sigma-1-it}) = x^{-sigma-1}cos(t log x), </p>
<p>and </p>
<p>Im(x^{-_sigma-1-it}) = -x^{-sigma - 1}sin(t log x).
</p>Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8504114
Tue, 22 Nov 2022 22:18:39 +0000Economist8504114@https://www.econjobrumors.com/<p>So are we settled that this is a gap?
</p>Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8498608
Sun, 20 Nov 2022 23:23:06 +0000Economist8498608@https://www.econjobrumors.com/<p>Then I'm not sure my message got across. If your post should say:</p>
<blockquote><p>
alpha_1(sigma+it', t) = alpha_1(sigma, t + t') = alpha_0(sigma+i(t+t'))
</p></blockquote>
<p>Then this is false (for \alpha_1 defined as in my 2) from the post at the top of this page). Very simply, for t fixed, this formula would imply that the non-constant holomorphic function z \mapsto alpha_1(z,t) takes only real values, and this is impossible. </p>
<blockquote><p>In fact, we could actually bound alpha/{1}(sigma + it) for complex sigma, directly from (18), by the usual methods in the literature (of bounding alpha).
</p></blockquote>
<p>Note that formula (18) describes a real-valued function, not \alpha_1 as considered in my point 2) above. So I don't understand your proposal.
</p>Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8496193
Sun, 20 Nov 2022 03:42:45 +0000Economist8496193@https://www.econjobrumors.com/<p>In fact, we could actually bound alpha/{1}(sigma + it) for complex sigma, directly from (18), by the usual methods in the literature (of bounding alpha).
</p>Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8496171
Sun, 20 Nov 2022 03:34:22 +0000Economist8496171@https://www.econjobrumors.com/<blockquote><p>Is that supposed to be \alpha_1 in place of \alpha on the right hand side?
</p></blockquote>
<p>Oh, yes...typo.
</p>Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8495864
Sun, 20 Nov 2022 01:26:41 +0000Economist8495864@https://www.econjobrumors.com/<p>Is that supposed to be \alpha_1 in place of \alpha on the right hand side?
</p>Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8495849
Sun, 20 Nov 2022 01:17:09 +0000Economist8495849@https://www.econjobrumors.com/<blockquote><blockquote>I mean, if sigma is fixed and complex so that sigma = sigma' + it', then<br />
s = sigma + it = sigma' + i(t+t')<br />
thus<br />
alpha_{1}(sigma + i(t + t')).<br />
Thus, estimating alpha_{1}(sigma + it) as t tends to infinity for fixed sigma in C, is essentially similar to estimating the quantity for real sigma.
</p></blockquote>
<p>This is my objection. It's not true that \alpha_1(\sigma+it,t') = \alpha_1(\sigma,t+t') -- replace \alpha_0 by a function like z -> |z|^2 to see this (as I wrote explicitly earlier).
</p></blockquote>
<p>Except that in this case, we indeed have </p>
<p>alpha_1}(sigma, t) = alpha(sigma, t + t') </p>
<p>since alpha_1 is holomorphic. It's bound is already what we have for alpha_0.
</p>Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8495688
Sun, 20 Nov 2022 00:14:58 +0000Economist8495688@https://www.econjobrumors.com/<blockquote><p>I mean, if sigma is fixed and complex so that sigma = sigma' + it', then<br />
s = sigma + it = sigma' + i(t+t')<br />
thus<br />
alpha_{1}(sigma + i(t + t')).<br />
Thus, estimating alpha_{1}(sigma + it) as t tends to infinity for fixed sigma in C, is essentially similar to estimating the quantity for real sigma.
</p></blockquote>
<p>This is my objection. It's not true that \alpha_1(\sigma+it,t') = \alpha_1(\sigma,t+t') -- replace \alpha_0 by a function like z -> |z|^2 to see this (as I wrote explicitly earlier).
</p>Economist on "A proof of the Riemann Hypothesis on the horizon."
https://www.econjobrumors.com/topic/a-proof-of-the-riemann-hypothesis-on-the-horizon/page/59#post-8495579
Sat, 19 Nov 2022 23:19:15 +0000Economist8495579@https://www.econjobrumors.com/<p>Alternatively, you may forget about everything I said, and try to estimate alpha_{1}(sigma + it) yourself, for sigma in C. I don't think this should be too hard for you.</p>
<p>It should help to recall that alpha_{1}(sigma + it) is the holomorphic continuation of alpha_{0}(sigma + it) to Re(sis) > Theta. So you just want to estimate alpha_{0}(sigma + it) for sigma in C.
</p>