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 separated by 2πi, depending on whether the integration is taken through complex values with positive or negative argument, and becomes infinitely small, for the former path, when the coefficient of i in the value of β becomes infinitely positive, but for the latter, when this coefficient becomes infinitely negative"
and I have no idea what he is saying.
What does "the integral from $0$ to x takes on values separated by 2πi, depending on whether the integration is taken through complex values with positive or negative argument" mean? Does "values separated by 2πi" mean to divide the integrand with $2\pi i$? And "depending on..." means divide the integrand if integral is taken through positive argument-complex value?
The integral's contour is on real number, but what does "depending on whether the integration is taken through complex values with positive or negative argument" mean? Which complex value is he talking about?
Fianally, even if I follow his instruction, how can I determine C by setting $\Im(\beta)=\pm \infty$?