I am looking for a series $\sum_{n \in \mathbb{Z}}a_ne^{int}$ with the following properties :
$\bullet a_0=0$
$\bullet a_n=a_{-n} \geq 0$ for all $n$ in $\mathbb{Z}$
$\bullet \sum_{n \in \mathbb{Z}}a_n < +\infty$
$\bullet \sum_{n \in \mathbb{Z}}|n|a_n = +\infty$
$\bullet$ It is easy to prove that $\sum_{n \in \mathbb{Z}}a_ne^{int} \sim1+r(t)$ with $t/r(t)\rightarrow 0$ near $0$
Does somebody have an idea of a series which satisfy those, or a proof of its existence which is not too hard or too long?
I have done my own research and am pretty sure I have found an elementary example : $a_n=\frac{1}{|n|^b}$ with $1<b<2$ whenever $n \neq 0$. This satisfies everything except the "easiness" condition for the asymptotics computation, as it involves Mellin Transform and residues... which is tedious. I would like to eschew using high-powered results as much as possible.
I have tried computing Fourier coefficients of functions satisfying the asymptotics near $0$ but have not found anything satisfactory so far. I would like the easiest example you can think of.
Thank you for your help!
If you take for granted the fact that
$$ \sum\limits_{n=1}^\infty \frac{1}{n^2}= \frac{\pi^2}{6}, $$
choosing
$$ a_n= \begin{cases} 0 &; n=0 \\ \frac{3}{\pi^2 n^2} & ; n\neq 0 \end{cases} $$
satisfies all but the "easiness" condition that you mentioned. I'm not sure how you verify that condition, but maybe it would apply here?