In Stein and Shakarchi's Fourier Analysis book, I am working Problem 1 of Chapter 3, where we are walked through a proof that $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n^\alpha}$ is not the Fourier Series of any function $f\in L^2[-\pi,\pi]$ $\hspace{2mm}\forall 0<\alpha<1$, despite the fact that the sum converges pointwise everywhere on the interval. In parts (a) and (b), we introduce the conjugate Dirichlet Kernel $$\tilde{D}_N (x) = \sum_{|n|\leq N} \text{sgn}(n) e^{inx}$$
We then show that if $f$ is Riemann integrable on $[-\pi,\pi]$, then $\int_{-\pi}^{\pi}|f(x) \tilde{D}_N (x)|dx \leq \mathcal{O}(\text{log}(N))$.
Part (c) asks us to consider the function $g=\sum_{n=1}^{\infty} \frac{\sin(nx)}{n^\alpha}$, and show that if we assume it is the Fourier Series of some Riemann integrable function $f$, this leads to the contradiction that $\sum_{n=1}^{N}\frac{1}{n^\alpha} = \mathcal{O}(\text{log}(N))$.
The book clearly wants me to plug in the function we are given to part (b), and this is where I have my questions.
(1) We know g converges pointwise on $[-\pi,\pi]$, but is it integrable? I assume it is, but how do we know this?
(2) If g is integrable, we can take its Fourier series. Is the Fourier Series of g also $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n^\alpha}$? How could we show this?