I need some clarification on the following, if possible: I have seen in that for every $ f \in S_k$ which Fourier transform is $\sum_{n=1}^\infty a(n)q^n$ there is an upper bound $\sum_{n=1}^N \|a(n)\| \leq c_f \cdot N^{\frac{k+1}{2}} $.
Now, somehow, using the theorem that states $ \| \sum_{n=1}^N a(n) \| \leq c_f \cdot N^{\frac{k}{2}} \cdot logN $ we've got that there are a lot of sign changes in the coefficients. I can't see how it follows, as it wasn't stated anywhere that the first bound is tight.
Can someone explain how this result is obtained? And maybe give me some reference to literature on the topic (if exists), of advanced undergraduate - beginner graduate level?
Thanks in advance.
In lieu of another answer, I'd wager that the Iwaniec-Kowalski book "Analytic Number Theory" (AMS) would provide you with prototypes for many such arguments.