We give an asymptotic formula for correlations

$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$

$f,\ldots ,f_{m}$

$f:\mathbb{N}\rightarrow \{-1,+1\}$

$1957$

$f(n)=n^{s}$

$\operatorname{Re}(s)<1$

$|f(n)|$

$n=a+b$

$a,b$

$\mathbb{N}$

In this paper, we study how small a box contains at least two points from a modular hyperbola $xy\equiv c\;(\text{mod}\;p)$ . There are two such points in a square of side length $p^{1/4+\unicode[STIX]{x1D716}}$ . Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6+\unicode[STIX]{x1D716}}$ or the least quadratic non-residue is less than $p^{1/(6\sqrt{e})+\unicode[STIX]{x1D716}}$ .

We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known methods do not apply. In particular, in the case of multiplicative characters modulo a prime $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$ we break the barrier of $p^{1/4}$ for ranges of individual variables.

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.

It is shown that the method of estimation of exponential sums with nonlinear recurring sequences, invented by the authors in a recent series of works, can be applied to estimating sums of multiplicative characters as well. As a consequence, results are obtained about the distribution of power residues and primitive elements in such sequences.