During his investigation of zeta Riemann defined the $\xi$ function as $\xi(s):= \Gamma(\frac{s}{2})(s-1)\pi^{-s/2}\zeta(s)$ which is an entire function that is invariant under the substitution $s \to 1-s$. Moreover $\xi$ shares its zeros with Riemann zeta function $\zeta$.


Riemann wanted to write $\xi(s)$ in the form $\xi(0)\prod\_{\rho}(1-\frac{s}{\rho})$ where the product is taken over all zeros of the zeta function. How does one prove the convergence of this product?



Convergence of Riemann's Product representation of Xi
You need to group the complex conjugates pairs of non-trivial zeros together


$$2\sum\_{\rho} \log(1-\frac{s}{\rho})=
2\sum\_{\Im(\rho)\le 2|s|} \log(1-\frac{s}{\rho})+
2\sum\_{\Im(\rho)> 2|s|}\log(1-\frac{s}{\rho})+\log(1-\frac{s}{\overline{\rho}})$$
$$=2\sum\_{\Im(\rho)\le 2|s|} \log(1-\frac{s}{\rho})+
2\sum\_{\Im(\rho)> 2|s|}O(\frac{|s|+|s^2|}{\Im(\rho)^2})\tag{1}$$
Due to the density of zeros theorem $$\# \{\rho, |\Im(\rho)|<T\}=O(T\log T)$$
$(1)$ converges locally uniformly, with a rate of convergence similar to $\sum\_{n>2|s|\log |s|} \frac{(|s|+|s^2|) \log^2 n}{n^2 }$.


