Question

The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series

{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }
Leonhard Euler already considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem. He also proved that it equals the Euler product

{\displaystyle \zeta (s)=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdot {\frac {1}{1-11^{-s}}}\cdots }{\displaystyle \zeta (s)=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdot {\frac {1}{1-11^{-s}}}\cdots }
where the infinite product extends over all prime numbers p.[1]

The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is valid for all complex s. This is permissible because the zeta function is meromorphic, so its analytic continuation is guaranteed to be unique and functional forms equivalent over their domains. One begins by showing that the zeta function and the Dirichlet eta function satisfy the relation

{\displaystyle \left(1-{\frac {2}{2^{s}}}\right)\zeta (s)=\eta (s)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}={\frac {1}{1^{s}}}-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-\cdots .}{\displaystyle \left(1-{\frac {2}{2^{s}}}\right)\zeta (s)=\eta (s)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}={\frac {1}{1^{s}}}-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-\cdots .}
But the series on the right converges not just when the real part of s is greater than one, but more generally whenever s has positive real part. Thus, this alternative series extends the zeta function from Re(s) > 1 to the larger domain Re(s) > 0, excluding the zeros {\displaystyle s=1+2\pi in/\ln(2)}s = 1 + 2\pi in/\ln(2) of {\displaystyle 1-2/2^{s}}1-2/2^s where {\displaystyle n}n is any nonzero integer (see Dirichlet eta function). The zeta function can be extended to these values too by taking limits, giving a finite value for all values of s with posit