Integral countour with Mangoldt function 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:

![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.