Confusion in proof for writing Poincares series in terms of Kloosterman sum

Question

I am currently self reading " Spectral theory of Riemann zeta function" by Yoichi Motohashi. The example is in first chapter and of poincare series. I am not able to justify from 1.1.5 to the next line. Actually I don't know what does mean by classifying summands according to h (mod l). Any help/Hint would be appreciated. I am stuck here since yesterday.

Answer 1

Since the summary has very often the expression $h/l$ , it is sensible to ask: how can I divide this fraction in integral part and fractional part? We can also hope, since some functions over there are periodic (e.g. $\exp( i\cdot) $) that things with the same fractional part will contribute in a similar fashion.

Hence we set $h = l n + h'$, with $h' = 1, \ldots, l $ and $n= - \infty, \ldots, +\infty$. The new $h'$ is the "classification $\pmod{l}$ " of the summand relative to $h$. Recall that $h$ has to be comprime with $l$, so that $ 1= (ln+h', l) = (h', l) $: the classification index has to be coprime with $l$. If $g(l, h) $ is your summand, you can then write

$$\sum_{l=1}^{\infty}\sum_{h=-\infty, (h, l) =1}^{+\infty} g(l, h) = \sum_{l=1}^{\infty} \sum_{h'=1}^{l} \sum_{n=-\infty}^{+\infty} g(l, nl+h') $$