Zeroes of a tricky function. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T12:35:11Z http://mathoverflow.net/feeds/question/46445 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46445/zeroes-of-a-tricky-function Zeroes of a tricky function. Alex Botros 2010-11-18T02:02:39Z 2010-11-18T05:20:29Z <p>I am attempting to show that there does not exist an N past which every open unit interval (k, k+1) -where k is an integer- contains a zero of the following function:</p> <p><code>$h(x)=\sum_{n=2}^{[\sqrt(x)]} \frac{\cot(x\pi/n)}{n}+\frac{\cot((x+2)\pi/n)}{n}$</code></p> <p>Where the <code>$[\sqrt(x)]$</code> is the lowest integer of the square root of <code>$x$</code>. Any thoughts? I had figured that I could consider the interval <code>$(i^2, (i+1)^2)$</code> in which h(x) is described by the function</p> <p><code>$h(x)=h_n(x)=\sum_{n=2}^{i} \frac{\cot(x\pi/n)}{n}+\frac{\cot((x+2)\pi/n)}{n}$</code></p> <p>Then try to see on what unit intervals (k, k+1) in here contain a zero of <code>$h_n$</code> then try to show that <code>$((i+1)^2, (i+2)^2)$</code> also has such an interval, but I am running out of ideas.</p>