From Norbert Wiener's proof of the PNT (with Tauberian theorem) there is a detail that I don't quite get. $\Lambda(n)$ is von Mangoldt's function.
Let
$$g(y) = \sum_{n=1}^{[e^y]}\frac{\Lambda(n)}{n}$$
$$(1)\hspace{10mm} \overline{\lim}_{N\to\infty}\frac{1}{N}\int_0^N \eta~ dg(\log \eta)= \overline{\lim}_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\Lambda(n)$$
I see that $g(\log \eta) = \sum_1^{\eta}\Lambda(n)/n$ and that l.h.s. (1) is a Stieltjes integral which, letting $f = \eta$ and keeping $g$ as is would be
$$\int \eta~ dg(\log \eta)=\int f ~dg = \int \eta~g'(\log \eta)d\eta$$ which could hopefully be done by parts but this does not seem to get me the right result--am left with an extra $n$ in the denominator--I just get $g(\log\eta)$ back...thanks for any assistance.
If $g$ were continuously differentiable, you'd get $$\eta \cdot \frac{d}{d\eta} g(\log \eta) = \eta \cdot g'(\log \eta) \cdot \frac{1}{\eta} = g'(\log \eta)\,.$$ But $g$ isn't continuously differentiable, so we integrate by parts. Using $g(\log \eta) = 0$ for $\eta < 2$ we have
\begin{align} \int_0^N \eta\,dg(\log \eta) &= \int_1^N \eta \,dg(\log \eta) \\ &= \eta g(\log \eta)\biggr\rvert_1^N - \int_1^N g(\log \eta)\,d\eta \\ &= Ng(\log N) - \int_1^N g(\log \eta)\,d\eta \\ &= Ng(\log N) - \int_1^N \sum_{n \leqslant \eta} \frac{\Lambda(n)}{n}\,d\eta \\ &= Ng(\log N) - \sum_{n\leqslant N} \frac{\Lambda(n)}{n} \int_1^N \chi_{[n,+\infty)}(\eta)\,d\eta \\ &= Ng(\log N) - \sum_{n\leqslant N} \frac{\Lambda(n)}{n}\cdot(N-n) \\ &= Ng(\log N) - N\underbrace{\sum_{n \leqslant N} \frac{\Lambda(n)}{n}}_{g(\log N)} + \sum_{n \leqslant N} \Lambda(n) \\ &= \sum_{n \leqslant N} \Lambda(n)\,. \end{align}