Wikipedia:Reference desk/Archives/Mathematics/2017 April 2

= April 2 =

Non-Hausdorff totally disconnected space
Every totally disconnected space is T1. Is there an example of a space that is totally disconnected but not Hausdorff (T2)? GeoffreyT2000 ( talk,  contribs ) 02:37, 2 April 2017 (UTC)


 * According to Steen & Seebach, a modified Fort space is just such a space. --Deacon Vorbis (talk) 04:12, 2 April 2017 (UTC)

Intuition Behind the Riemann &zeta; Functional Equation
Let $$~\rho(x)~=~\dfrac{\Gamma(x)\cdot\zeta(2x)}{\pi^x}~.\quad$$ Then $$~\rho\Big(\tfrac14+x\Big)~=~\rho\Big(\tfrac14-x\Big)~$$ constitutes the functional equation for the Riemann $$\zeta$$ function. The presence of the product $$~\Gamma(x)\cdot\zeta(2x)~$$ is perfectly understandable, inasmuch as the poles of the former coincide with the (trivial) zeroes of the latter; as is also the symmetry with regard to $$\tfrac14,$$ since this value stands midway between &rho;'s two poles. What poses serious difficulties from an intuitive perspective, however, is the presence of $$~\dfrac{\zeta(2x)}{\pi^x}~$$ instead of the expected $$~\dfrac{\zeta(2x)}{\pi^{\color{red}2x}}~,~$$ given the fact that $$\zeta(2k)$$ always possesses a known closed form in terms of $$\pi^{2k}$$ rather than merely $$\pi^k$$ for integer values of the argument k. One can, of course, always write $$~\rho(x/2)~=~\dfrac{\Gamma(x/2)~}{~\Gamma(1/2)^x}\cdot\zeta(x),~$$ but, for all its niceness, the latter appears somewhat contrived, inasmuch as the power of $$\pi$$ is clearly a contribution of Riemann's $$\zeta$$ rather than Euler's $$\Gamma$$ function. — 79.113.203.86 (talk) 14:29, 2 April 2017 (UTC)