Proof analytic of prime number theorem

Question

In the proof of analytic prime number theorem

how can i justify

$\int_{m}^{m+1} \sum_{n \leq x} \Lambda (n) dx = \sum_{n \leq m} \Lambda (n)$

where $\Lambda$ is Mangoldt function

Answer 1

$\sum_{n \leq x} \Lambda (n)$ is constant on any interval not containing an integer, and its value on the interval $(m,m+1)$ equals $\sum_{n \leq m} \Lambda (n)$.