On the asymptotic distribution of values of arithmetic functions. (Q1432502)

This paper is concerned with problems related to the distribution of values of incomplete\break Kloosterman sums with prime numbers modulo primes, incomplete Gauss sums and their generalizations, and the joint distribution of quadratic residues and nonresidues modulo different primes. Six theorems are announed of which the following states an asymptotic distribution law for the values of incomplete Gaussian sums. Theorem 3. Let \(\chi\) be a complex-valued Dirichlet character. Put \[ S_h(x)= \sum_{n=x+1}^{x+h} \chi(n) e^{2\pi i \frac {an}{p}}, \] where \(p\) is a prime, \((a,p)=1\), and the numbers \(x\) and \(h\) are integers such that \(0\leq x< p\) and \(0< h< p\). Then, as \(p\to \infty\), \(h= h(p)\to \infty\), and \(\frac {\log h}{\log p}\to 0\), the value \(\xi=| \frac {S_h(x)} {\sqrt{h}} |^2\) has asymptotically an exponential distribution with parameter 1; i.e., at every fixed \(y> 0\), we have \[ \frac 1p N_p \{\xi< y\}\to \int_0^y e^{-v}\, dv. \] Using the properties of the Gauss sums and the method of Vinogradov, the authors obtain an asymptotic formula for the umber of quadratic residues and nonresidues in certain sequences modulo different numbers. Finally, they state an analogue of the central limit theorem for the distribution of the values of sums of Legendre symbols modulo different numbers.