In an analytic proof for prime number theorem I found a passage I could not understand:
$$A=\sum_{n\leq x} \frac{1}{2 \pi i} \int_{c-\infty i}^{c+\infty i} \frac{\Lambda(n) (x/n)^s ds}{s (s+1)} = \sum_{n=1}^{\infty} \frac{1}{2 \pi i} \int_{c-\infty i}^{c+\infty i} \frac{\Lambda(n) (x/n)^s ds}{s (s+1)}$$ where $c>1$, $x>1$ and $$A= \psi_1(x)/x = \sum_{n\leq x} (1-n/x) \Lambda(n).$$
The book say that integral vanishes if $x < n$. Someone understand why?
The reason is that the integral $$\int_{c-i\infty}^{c+i\infty}\frac{x^s}{s(s+1)}\,ds$$ vanishes when $x<1$. To prove that, consider the contour:
(image is from Apostol's Introduction to Analytic Number Theory, where you can also find the details of the proof). Elementary estimates shows that the integral over the arc tends to zero as $R\to\infty$, so Cauchy's theorem implies that the above integral is zero since the function has no poles inside the contour.