I found the following in a book and don't understand. Let $\chi$ denote a non-principal character modulo $q$ and $S(x)=\sum_{n\leq x}\chi (n)$. Then
$\sum_{m>y} \frac{\chi(m)}{m} = \int_y^{\infty }u^{-1}dS(u)=-\frac{S(y)}{y} + \int_y^{\infty }S(u)u^{-2}du$
What does "$dS(u)$" mean?
In Apostol's book on analytic number theory this is Abel's identity applied to the function $f(t)=\frac{1}{t}$, i.e., $$ \sum_{x<m\le y}\frac{\chi(m)}{m}=\frac{S(y)}{y}-\frac{S(x)}{x}+\int_x^y \frac{S(t)}{t^2}dt, $$ which gives for $y\to \infty$ the result $$ \sum_{m\le x} \frac{\chi(m)}{m}=\sum_{m=1}^{\infty} \frac{\chi(m)}{m}+O(1/x). $$ Here the integral is the usual Riemann integral, but I also think that yours is the Riemann-Stieltjes integral, as Jyrki said. If you tell us which book you mean, then one could verify this of course.