On New Metric Theorems in Probabilistics Number Theory

The present work is a current research on the interface between analytic number theory, probability theoryand mathematical statistics. The aim of the work is a systematic study of the distribution laws of the valuesof various normalized short sums, that is to say sums with numerical functions, the length of the summationinterval is small compared to the value of their period, or in other words in comparison with a length of thecorresponding full sums. The first results in this field belong to renowned authors of number theory: G.Davenport and P. Erdös who 80 years ago demonstrated that under certain conditions, applied to the values ofthe Legendrian modulus and the length of their sums short normed, are asymptotically distributed according toa normal or exponential law. In the USSR, this theme found its extension in a main article by I. P. Koubiliousand U. Linnik on the arithmetic modeling of Brownian movements. In the Moscow school of analytic numbertheory, this area was actively developed in the works of Postnikov and M. P. Miniev. A new intensive splash ofresearch in this area of number theory was initiated in the 90s by V. N. Shubarikov.