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...

In Riemann's paper, he calculated $${1\over 2\pi i\log x}\int_{a-\infty i}^{a+\infty i} x^s {d\over ds}{1\over s}\log(1-{s\over \beta})ds=\int_0^x {t^{\beta-1}\over\log t }dt+C$$ for $\Re(\beta)>0$ and he set the value of beta to find C. He said " the integral from $0$ to x takes on values separa...