In Zygmund's Trigonometric Series, vol I, on page 19 section 2.22 they write that Riemann showed that the Fourier coeff of $f(x)=\frac{d}{dx}(x^{\nu} \cos(\frac{1}{x}))$ for $\nu \in (0,\frac{1}{2})$
doesn't converge to zero, nor is it $o(n^{\frac{1-2\nu}{4}})$.
Well, the original paper is here: https://eudml.org/doc/203787
On the bottom of page 42 Riemann argues that the coeffecient equals:
$$\frac{1}{2}\sin(2\sqrt{n}-na+\pi/4)\sqrt{\pi} n^{\frac{1-2\nu}{4}}$$
The derivation of this identity is on page 44, where $\phi(x)=x^{\nu}$ and $\psi(x)=\frac{1}{x}$.
My obstacle here (besides that I don't read German, maybe in 2014-2015 I'll take a course in German to aid me with reading German articles) is that I don't understand the eqaution on the top of page 44.
I understand that he makes a change of variables: $dy/\sqrt{y-\beta} = \sqrt{2\psi''(\alpha)} dx$.
But I don't understand how did the limits of integration here change, and why did arguemtent of $\psi(x)$ and $\phi(x)$ changed to $\alpha$, where obviously $x^2 = \frac{2(y-\beta)}{\psi''(\alpha)}$ .
Any feedback is appreciated.