Vinogradov's method of estimation of exponential sums over primes allowed him to solve the number of arithmetic problems with primes.
One of them is a problem of distribution of the values of non-principal character on the sequence of shifted primes. In 1938 he proved that
if $q$ is an odd prime, $(l, q)=1$, $\chi (a)$ is non-principal character modulo $q$,
then
\begin{equation}
T(\chi )=\sum_{p\le x}\chi (p-l)\ll x^{1+\varepsilon} \left(\sqrt{\frac{1}{q}+\frac{q}{x}} +x^{-\frac{1}{6}}\right). \tag{IMV}
\end{equation}
This estimate is non-trivial when $x\gg q^{1+\varepsilon}$ and an asymptotic formula for the the number of quadratic residues (non-residues) modulo $q$ of the form $p-l$, $p\le x$ follows from it.
Later in 1953, I. M. Vinogradov obtained a non-trivial estimate of $T(\chi )$ when $x\ge q^{0,75+\varepsilon}$, $q$ is a prime. It was a surprising result. In fact, $T(\chi )$ can be represented as a sum over zeroes of correspondent Dirichlet $L$ — function; So a non-trivial estimate of $T(\chi )$ is obtained only for $x~\ge~q^{1+\varepsilon}$ provided that the extended Riemann hypothesis is true.
In 1968 A. A. Karatsuba found a method that allowed him to obtain non-trivial estimate of short sums of characters in finite fields with fixed degree. In 1970 using the modification of his technique coupled with Vinogradov's method he proved that: if $q$ is a prime number, $\chi$ is non-principal character modulo $q$ and $x\ge q^{\frac{1}{2}+\varepsilon}$, then the following estimate is true
$$
T(\chi )\ll xq^{-\frac{1}{1024}\varepsilon^2}.
$$ In 1985 Z. Kh. Rakhmonov generalized the estimate (IMV) for the case of composite modulo and proved:
let $D$ is a sufficiently large positive integer, $\chi$ is a non-principal character modulo $D$, $\chi_q$ is primitive character generated by character $\chi$, then
$$
T(\chi )\le x\ln^5x \left(\sqrt{\frac{1}{q}+\frac{q}{x}\tau^2(q_1)} +x^{-\frac{1}{6}}\tau (q_1)\right),
\qquad q_1={\genfrac{}{}{0pt}{}{p\backslash D}{p\not\backslash q}}p.
$$
If a character $\chi$ coincides with it generating primitive character $\chi_q$, then the last estimate is non-trivial for $x>q(\ln q)^{13}$.
In 2010 г. J. B. Friedlander, K. Gong, I. E. Shparlinski showed that a non-trivial estimate of the sum $T(\chi_q )$ exists for composite $q$ when $x$ — length of the sum, is of smaller order than $q$.
They proved: for a primitive character $\chi_q$ and an arbitrary $\varepsilon >0$ there exists such $\delta >0$ that for all $x\ge q^{\frac{8}{9}+\varepsilon}$
the following estimate holds:
$$
T(\chi_q )\ll xq^{-\delta}.
$$
In 2013 Z. Kh. Rakhmonov obtained a non-trivial estimate of $T(\chi_q)$ for the composite modulo $q$
and primitive character $\chi_q$ when $x\ge q^{\frac{5}{6}+\varepsilon}$.
In this paper the theorem about the estimate of the sum $T(\chi_q)$ is proved for cubefree modulo
$q$. It is non-trivial when $x\ge q^{\frac{5}{6}+\varepsilon}$.
Bibliography: 15 titles.
