The paper analyzes the number of zeros in the binary representation of a natural number. The analysis is carried out using the concept of the fractional part of a number, which naturally arises when finding a binary representation. This idea relies on the fundamental property of the Riemann zeta function, which is constructed using the fractional part of a number. Understanding that the ratio of the fractional and integer parts, by analogy with the Riemann zeta function, expresses the deep laws of numbers, will explain the essence of this work. For the Syracuse sequence of numbers that appears in the Collatz conjecture, we use a binary representation that allows us to obtain a uniform estimate for all terms of the series, and this estimate depends only on the initial term of the Syracuse sequence. This estimate immediately leads to the solution of the Collatz conjecture