On a special trigonometric series and its applications, as well as the memories

The talk is devoted to the common results of the speaker and G.I. Arkhipov concerning the behavior of the series of the type
$$ h(P)\,=\,\sum\limits_{n\ne 0}\frac{e^{2\pi iP(n)}}{n}, $$
and of their symmetric partial sums
$$ h_{N}(P)\,=\,\sum\limits_{1\leqslant |n|\leqslant N}\frac{e^{2\pi iP(n)}}{n}, $$
where
$$ P(x)\,=\,P(x;\boldsymbol{\alpha})\,=\,\alpha_{1}x+\alpha_{2}x^{2}\,+\ldots\,+\alpha_{r}x^{r} $$
is a polynomial of degree $r\geqslant 1$ with real coefficients.