20 Riemann zeta function notes

\(\blacksquare \) Given by \(\zeta \left ( s\right ) =\sum _{n=1}^{\infty }\frac {1}{n^{s}}\) for \(\operatorname {Re}\left ( s\right ) >1\). Euler studied this and It was extended to the whole complex plane by Riemann. So the Riemann zeta function refer to the one with the extension to the whole complex plane. Euler only looked at it on the real line. It  has pole at \(s=1\). Has trivial zeros at \(s=-2,-4,-6,\cdots \) and all its non trivial zeros are inside the critical strip \(0<s<1\) and they all lie on the critical line \(s=\frac {1}{2}\). \(\zeta \left ( s\right ) \) is also defined by integral formula

\[ \zeta \left ( s\right ) =\frac {1}{\Gamma \left ( s\right ) }\int _{0}^{\infty }\frac {1}{e^{t}-1}\frac {t^{s}}{t}dt\qquad \operatorname {Re}\left ( s\right ) >1 \]

\(\blacksquare \) The connection between \(\zeta \left ( s\right ) \) prime numbers is given by the Euler product formula

\begin{align*} \zeta \left ( s\right ) & =\Pi _{p}\frac {1}{1-p^{-s}}\\ & =\left ( \frac {1}{1-2^{-s}}\right ) \left ( \frac {1}{1-3^{-s}}\right ) \left ( \frac {1}{1-5^{-s}}\right ) \left ( \frac {1}{1-7^{-s}}\right ) \cdots \\ & =\left ( \frac {1}{1-\frac {1}{2^{s}}}\right ) \left ( \frac {1}{1-\frac {1}{3^{s}}}\right ) \left ( \frac {1}{1-\frac {1}{5^{s}}}\right ) \left ( \frac {1}{1-\frac {1}{7^{s}}}\right ) \cdots \\ & =\left ( \frac {2^{s}}{2^{s}-1}\right ) \left ( \frac {3^{s}}{3^{s}-1}\right ) \left ( \frac {5^{s}}{5^{s}-1}\right ) \left ( \frac {7^{s}}{7^{s}-1}\right ) \cdots \end{align*}

\(\blacksquare \) \(\zeta \left ( s\right ) \) functional equation is

\[ \zeta \left ( s\right ) =2^{s}\pi ^{s-1}\sin \left ( \frac {\pi s}{2}\right ) \Gamma \left ( 1-s\right ) \zeta \left ( 1-s\right ) \]