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

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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 ζ Functional Equation

Let $~\rho (x)~=~{\dfrac {\Gamma (x)\cdot \zeta (2x)}{\pi ^{x}}}~.\quad$ Then $~\rho {\Big (}{\tfrac {1}{4}}+x{\Big )}~=~\rho {\Big (}{\tfrac {1}{4}}-x{\Big )}~$ 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 ${\tfrac {1}{4}},$ since this value stands midway between ρ'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)