The Riemann zeta function, is the function of the complex variable $s$, defined in the half plane $\Re(s)>1$ by the absolutely convergent series $\zeta(s) = \sum_{n} n^{-s}$

and extends to the whole of $\mathbb{C}$ by analytic continuation. It is known that all the complex zeros of $\zeta(s)$ satisfy $0<\Re(s) <1$.

But am not sure how on they are calculated ?


Basically, naming $s=\sigma+it$, the idea is to use the function $$ \xi(s)=\Gamma(s/2)\pi^{-s/2}(s-1)\zeta(s) $$ because it's real-valued on the critical line $t=1/2$, hence you'll find a zero whenever $\xi(1/2+it)$ changes sign. There are various method to do that, a very nice introduction can be found in Edwards' book ``Riemann's Zeta Function", see for example Section 6.5.

  • $\begingroup$ Thank you @Pittaluga. I understand that the R.H can also be stated as ''all zeros of $\xi(t)$ are real". Since both factors $s-1$ and $\pi^{-s/2}$ are nonzero, it seems the only factor that can be zero is the gamma function ? $\endgroup$ – User1 Feb 8 '16 at 16:40
  • $\begingroup$ It's not so easy, you're dealing with complex numbers, so you have to take into account the arguments...take a look at the reference I suggested you, it's really well written, mostly undergraduate level! $\endgroup$ – PITTALUGA Feb 8 '16 at 19:39

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