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I encounter a tricky sum like the Kloosterman sum

$${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$ where $l$ is a positive integer co-prime with $P$ and here $P$ is a prime number, $a$ is a positive integer co-prime with $P$. $\overline{x}$ means $x\overline{x}≡1(\mod P)$.

If this sum can be bounded by sqauare-root $O(P^{\frac{1}{2}+\varepsilon})$? (which follows by invoking Weil bound for Kloosterman sums or some knowledge on exponential sums estimates developed by some experts like A. Adolphson, S. Sperber, E. Bombieri, N. Katz which part I am not familiar with, to be frankly) Did anyone ever saw this kind of sum before with a square module $P^2$ at the tail? Please share some comments. Many thanks.

Note that one has the estimate for the exponential sum: for $(a,p)=1$, and $\chi$ a character modulo $P$ $${\sum_{x \mod P}}^\ast \chi(x)e\left(\frac{ax+\overline{x}}{P}\right)\ll P^{\frac{1}{2}}.$$ In our case the exponential factor is of modulo $P^2$. By now I haven't fond any reference using the sophisticated knowledge on exponential sum in finite field to get a power-saving bound (noting the completing method would give the trivial bound $O(P)$). If any expert here have some strategies or references, please give a guide.

Your any opinions are highly appreciated. Thanks in advance.

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    $\begingroup$ Your sum is not correctly defined. For summand $f(x)$ we have $f(x)\ne f(x+P).$ $\endgroup$ – Alexey Ustinov Jan 15 '18 at 9:49
  • $\begingroup$ @ Alexey Ustinov It looks more like a Kloosterman sum twisted with the harmonics $e\left(\frac{l x^2}{P^2}\right)$ which is in-complete, rather than a Gauss sum. $\endgroup$ – Fei Jan 15 '18 at 9:58
  • $\begingroup$ Usually notation $x\mod P$ means that the sum is full, i.e. $f(x)=f(x+P)$. Otherwise it is not clear what is $x\mod P$. $\endgroup$ – Alexey Ustinov Jan 15 '18 at 10:07
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    $\begingroup$ I am not sure you understood Alexey Ustinov's comment. You are summing an expression that is not periodic modulo $P$. That is, your sum depends on which representatives you choose for your residue classes. If you do not give specify the set of representatives, your sum (hence your question) makes no sense. $\endgroup$ – GH from MO Jan 15 '18 at 21:35
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    $\begingroup$ @Fei: Once again, your original sum, and also the sum in your previous comment make no sense. There is no $e(ax^2/P^2)$ for a residue class $x\bmod P$. There is $e(ax^2/P^2)$ for $x=p-7$ and also for $x=-7$, but these are not the same! $\endgroup$ – GH from MO Jan 17 '18 at 6:01

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