While studying analytic number theory from Tom M Apostol introduction to analytic number theory I am unable to think about a argument used in Theorem 13.4 .
Image highlighting the line in which I have doubt -
My doubt is 1.why Apostol makes N depend on t and 2. why he is not violating generality in doing so?
Can someone please tell.
We make $N$ depend on $t$ because there are two free variables in equation (17), $N$ and $s$, but the target inequalities (19) and (20) depend on $t$ only. So we are going to make $N$ and $s$ depend on $t$ to get results depending on $t$ only. The first is $|s| < 2t$. You are asking about the second.
Equation (17) is from theorem 12.21. Theorem 12.21 only requires $N$ is a non-negative integer. The choice $N = \lfloor t \rfloor$ makes $N$ a non-negative integer (because $t \geq \mathrm{e}$), so is consistent with the hypotheses of theorem 12.21. That is, the equation is valid for any (non-negative integral) choice of $N$, including the one made in this proof.