User barry brent - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T02:41:58Z http://mathoverflow.net/feeds/user/24247 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98909/solutions-of-zetas-1-zeta-zetas-1-near-a-line-and-a-circle-resp Solutions of $\zeta(s) = 1$, $\zeta(\zeta(s)) = 1$ near a line and a circle, respectively? Barry Brent 2012-06-05T21:21:32Z 2012-06-07T08:27:09Z <p>Are solutions of $\zeta(s) = 1$ very near a line $\Re(s) = 54$ and solutions of $\zeta(\zeta(s)) = 1$ either on or very near a circle with center $\approx .00936$ tangent to $\Re(s) = 1$, known to exist? I seem to be observing these things in computer experiments. The solutions to the first equation (zeta images of the solutions to the second one) look like they lie on circles of radius $\approx 4 \times 10^{-10}$ centered on points of the form $54 + i k \pi/\log(2), k$ odd.</p> <p>Is this ridiculous? What might produce such an illusion? </p> <p>Edit: To illustrate, here is a link to a PDF with graphics of the phenomenon: <a href="http://barrybrent.9f.com/zeta=1.pdf" rel="nofollow">http://barrybrent.9f.com/zeta=1.pdf</a>. (~ a meg.)</p> <p>There are other such apparent circles for higher zeta iterates, which I'll show in a later draft, if the observations aren't knocked down.</p> <p>Barry Brent</p> http://mathoverflow.net/questions/98909/solutions-of-zetas-1-zeta-zetas-1-near-a-line-and-a-circle-resp/98934#98934 Comment by Barry Brent Barry Brent 2012-06-06T21:03:33Z 2012-06-06T21:03:33Z By illusion I meant 'optical illusion'. I was asking, were my observations ruled out by theory. Circles, or curves very near circles, appear in the PDF graphics. I see above that Juan has answered me saying that the phenomenon actually is ruled out. You're right though in the sense that precision issues at 'microscopic' scales in my plots may have caused a visual illusion. Barry http://mathoverflow.net/questions/98909/solutions-of-zetas-1-zeta-zetas-1-near-a-line-and-a-circle-resp/98977#98977 Comment by Barry Brent Barry Brent 2012-06-06T20:51:49Z 2012-06-06T20:51:49Z Juan, thanks. Barry http://mathoverflow.net/questions/98909/solutions-of-zetas-1-zeta-zetas-1-near-a-line-and-a-circle-resp/98934#98934 Comment by Barry Brent Barry Brent 2012-06-06T15:29:05Z 2012-06-06T15:29:05Z This isn't exactly what I was asking. The linked paper already mentions the precision issues in the discussion of $\Re(s) \approx 222.48$. In view of the graphics included in the PDF (linked above) the geometry of the graph of zeta, not the numerical estimates, suggested the proposal. So I asked, not whether it's true, but whether it's obviously wrong.