Varbanets S. Method on exponential sums in theory of congruential generators of the pseudorandom numbers and asymptotic problems in number theory

This thesis is devoted to investigation the generating problems of the sequences of pseudorandom numbers using a competitive recursion of the prime power modulus, as well as the problems of analytical number theory that arise with constructing the asymptotic formulas for summatory functions associated with the distribution of divisor functions τ_k, k = 2,3 over the rings of rational integers or Gaussian integers. We introduced the construction of new non-trivial estimates of purely completed or twisted exponential sums with a polynomial in the exponent over the ring of Gaussian integers. In addition, there are investigated the special exponential sums of Klosterman type over the ring of integers of an imaginary quadratic expansion of the field of rational numbers. The studied norm Klosterman sums have no analogue in the rational case, and their estimates are used to obtain the estimates of an error terms in problems of analytic number theory such as the problem of circle (or ellipse) in arithmetic progression and in the coding theory with Klosterman code problems etc. The obtained estimates of the norm Klosterman sums are related to the results of P. Deligne and E. Bombieri on the Riemann hypothesis for algebraic varieties. R. Evans, G. Perelmuter, S. Stepanov, R. Dabrovsky, V. Fischer, H. Ivanets and others were engaged in the development of methods for estimating of such sums. The significance of the obtained results on estimates of completed exponential sums is that the asymptotic formulas for estimates of the distribution of arithmetic functions on arithmetic progressions are based on such estimates. The second part of thesis is devoted to construction the inversive congruential generators modulo the power of prime rational number p. We gave the generalizations of the inversive congruential generator. We also investigated the inversive congruential generator of second order. Here also we constructed a new type of generators for which the relative recursion is based on the properties of elements from so-called the norm group, which is a subgroup of the multiplicative group of residue classes of the ring Z[θ] modulo the p^m. We used the constructed exponential sums to obtain the non-trivial estimates for discrepancy function of sequences of PRN’s. The obtained estimates of discrepant function improve the results of Niederreiter and Shparlinskii. The last part of thesis was being devoted to the problems of analytical number theory.