I am currently learning analytic number theory using Davenport's Multiplicative Number Theory book, and at some point I believe something silly is happening. I have great faith that I am wrong AND that I am right. Here's the thing.
At some point during the proof of Dirichlet's proof of the theorem about primes in arithmetic progressions, using Poisson's summation formula, one ends up with the integral $ \int_{-\infty}^{\infty} e^{2\pi i N y^2} \, dy. $ Here $N$ is an integer. This improper integral should, in practice, evaluate to $\frac{1+i}{2\sqrt N}$. Actually, the reason why we need to compute this is because we know that something we need is equal to $ \lim_{Y,Z \to \infty} \int_{-Y}^{Z+1} e^{2\pi i N t^2} \, dt, $ where $Y$ and $Z$ take integer values, and because of some Fourier series argument (which is not the purpose of my question), I know that this limit exists. The way Davenport computes this limit is by evaluating the above integral (not the limit one, the one above it). The way it is computed is by first proving that it converges, and then uses some identity which I don't have problems with. The argument Davenport uses is that for $Y' > Y > 0$, we have $ \int_Y^{Y'} e^{2\pi i N y^2} \, dy = \frac 12 \int_{Y^2}^{Y'^2} \frac{e^{2\pi i N z}}{\sqrt z} \, dz $ after the change of variables $z = y^2$, and "this is where I'm stuck, magic happens" : supposedly that "after using the second mean value theorem, or by integration by parts, this should have absolute value $O(\frac 1Y)$ as $Y \to \infty$". How is that? I know both integration by parts/second mean value theorems, but I have no idea how to get there ; naive applications of those two give me no big-oh at all ; for instance, $ \frac 12 \int_{Y^2}^{{Y'}^2} \frac{e^{2 \pi i N z}}{\sqrt z} \, dz = \frac 12 \left( \left. \frac{e^{2 \pi i Nz}}{2 \pi i N \sqrt z} \right|_{Y^2}^{{Y'}^2} + \frac 1{4\pi i N} \int_{Y^2}^{{Y'}^2} \frac{e^{2\pi i N z}}{(\sqrt z)^3} \, dz \right) $ The first term is $O(1/Y)$, but the second term, if I use the mean value theorem there's a $Y'$ in the numerator, which can get the thing arbitrary large ; what I want is that the integral from $Y$ to $\infty$ to be bounded for $Y$ large enough, so this is really annoying. I could integrate by parts again, but I would still get a $Y'$ in the numerator.
Another thing that annoys me is that is I write $ \int_{-\infty}^{\infty} e^{2\pi i N y^2} \, dy = \int_{-\infty}^{\infty} \cos(2\pi N y^2) + i \sin(2 \pi N y^2) \, dy, $
the real part and imaginary parts both don't seem to converge, since the tail doesn't go to zero, but oscillates very fast? I know that functions can oscillate and still be integrable in the Riemann-limit sense, but still, this looks suspicious.
Any explanations? Anything is welcome... thanks in advance!