While I was trying to translate Riemann's paper "On the numbers of prime less than a given quantity" into Korean, I've found a strage thing in it.
Riemann wrote on his paper $$\log\Xi(t)=\sum_{\Xi(\alpha)=0}\log(1- {t^2 \over \alpha^2} )+\log\Xi(0).$$
In fact, he used $\xi$ in stead of $\Xi$, but he defined in his paper $\xi$ as following $$\xi(t)=(s-1)\Gamma({s \over 2}+1)\pi^{-{s \over 2}}\zeta(s)$$ where $s= {1 \over 2}+it$, so I just use capital Xi.
Anyway, I already knew the Hadamard product about $\xi$ function, which is $$\xi(s)=\xi(0)\prod_{\rho}(1-{s \over \rho}).$$ but I've never heard of, or seen the product formula about $\Xi$ function. I googled about them, of course, but couldn't get anything. I also guessed that product formula is derived from $\xi$ function Hadamard product, but I was impossible to derive it. Since there was no proof about that product in the paper, so I have no idea how where it came from. Can anybody explain me how to prove it? Or did Riemann jsut use $\xi$ as $(s-1)\Gamma({s\over2}+1)\pi^{-{s\over2}}\zeta(s)$?
Also, since I'm not really good at English(I'm Korean student), I was not able to understand some terms. For example, Riemann said the integral $\int d\log\xi(t)$ is equal to $(T\log{T\over2\pi}-T)i$ up to a fraction of the order of magnitude of the quantity ${1\over T}$. Does that mean that $\int d\log\xi(t)=(T\log{T\over2\pi}-T)i+O({1\over T})$?