Upper bound of Riemann Zeta function

Question

In the book of Apostol Introduction to Analitic number theory at pag. 271 about $\zeta$ upper bound

$\sum_{n=1}^{N} \frac{1}{(n+a)^{\sigma}} \leq 1 + \int_{1}^{N}\frac{dx}{(x+a)^\sigma} $

with $0<\sigma<2$.

Does any know how this formula came from?

Answer 1

View the sum as the total area of rectangles with unit width and heights $\frac{1}{(n+a)^\sigma}$. This can be interpreted as a left Riemann sum for the integral on the right. (A small adjustment has been made to start the integral at $1$ instead of $0$ because the integral may diverge near $x=0$.) This visualization may be useful.