I am reading this paper. I am stuck on something, which I think is something basic but I haven't been able to figure it out yet, and I was hoping someone could explain it to me.

Let $$ \sigma(u) = \frac{1}{2\pi} \int_{-T}^T F(t) \frac{u^{1/2 + it} - M^{1/2 + it}}{1/2 + it} dt + O(N (\log N)^2/T), $$ where $F$ is a analytic function and $M$, $T$ are fixed positive numbers. Then on page 52 of the paper it is stated $$ \int_M^N e(u \lambda) d\sigma = \frac{1}{2\pi} \int_{-T}^T F(t) \int_M^N e(u \lambda) u^{-1/2 + it} \ du \ dt + E $$ where $$ E \ll (\log N)^2 (1 + |\lambda| N)N/T. $$ I was wondering about how I can show this bound on the error term. I would greatly appreciate any explanation. Thank you very much!

PS Here $e(z)$ denote $e^{2 \pi i z}$.