The estimates of short weighted Kloosterman sums

The incomplete weighted Kloosterman sum is the exponential sum of the type
$$ S(x, m;a, b) = \sum_{\substack{\nu \leqslant x\\ (\nu, m) = 1}}{ f(\nu )\exp\Big( 2\pi i \frac{a\overline{\nu} + b\nu}{m}\Big)}, $$
where $m, a, b$ are integers, $1<x<m$, the weight $f(\nu)$ is some arithmetic function and $\overline{\nu}$ denotes the inverse residue to $\nu$ modulo $m$, that is, $\nu \overline{\nu} \equiv 1 \pmod{m}$.

In the talk, we'll speak about some new estimates for incomplete Kloosterman sums with weights for case when $m \geqslant m_0$ is a prime, $(a,m)=1$ and the length $x$ of the sum lies in the interval
$$ \exp(c (\ln m)^{2/3} (\ln\ln m)^{4/3}) \leqslant x \leqslant \sqrt{m},\quad c>0. $$
The weight function $f$ here is the Möbius function, the characteristic function of the set of the square-free numbers, the multidimensional divisor function and the characteristic function of the set of numbers that are the sum of two squares of integers. Our estimates are based on the method developed by A.A. Karatsuba [1], [2] in 1990's, and they improve the previous result obtained by M.A. Korolev [3] in 2010.

[1] A.A. Karatsuba, Fractional parts of functions of a special form, Izv. Math., 59 (1995), № 4, p. 721–740.

[2] A.A. Karatsuba, Analogues of Kloosterman sums, Izv. Math., 59 (1995), № 5, p. 971–981.

[3] M.A. Korolev, Short Kloosterman sums with weights, Math. Notes, 88 (2010), № 3, p. 374–385.