The core motivation for this study emanates from Voronin S.'s seminal work on the universality of the zeta function. This universality positions the zeta function as a pivotal element in the conceptual framework of global artificial intelligence, suggesting its potential as a foundational component. A critical barrier to harnessing this promising universality has been the absence of efficient algorithms capable of actualizing this concept.

This paper presents an analysis of the number of zeros in the binary representation of natural numbers. The primary method of analysis involves the use of the concept of the fractional part of a number, which naturally emerges in the determination of binary representation. This idea is grounded in the fundamental property of the Riemann zeta function, constructed using the fractional part of a number. Understanding that the ratio between the fractional and integer parts of a number, analogous to the Riemann zeta function, reflects the profound laws of numbers becomes the key insight of this work. The findings suggest a new perspective on the interrelation between elementary properties of numbers and more complex mathematical concepts, potentially opening new directions in number theory and analysis.

We show how the Poincar\'e--Riemann--Hilbert boundary-value problem enables us to construct effective estimates of the potential in the Schr\"odinger equation. The apparatus of the three-dimensional inverse problem of quantum scattering theory is developed for this. It is shown that the unitary scattering operator can be studied as a solution of the Poincar\'e--Riemann--Hilbert boundary-value problem. This allows us to go on to study the potential in the Schr\"odinger equation

This study delves into the Schrödinger equation's new invariants, shedding light on their crucial implications for theoretical physics and beyond. Specifically, it explores the behavior of scattering amplitudes and bound states concerning the choice of coordinate systems. The research unveils that while the energy of bound states remains invariant under coordinate transformations, the phase of scattering amplitudes undergoes variations, underscoring its pivotal role in theories reliant on phase normalization. Moreover, these findings have played a pivotal role in constructing estimates for three-dimensional Navier-Stokes equations, enhancing our ability to model complex fluid dynamics with greater precision and reliability. Additionally, in seismic exploration, tomography, and ultrasound imaging, these properties allow researchers to optimize phase selection, facilitating the most effective interpretation of measurement results. This strategic phase manipulation ensures clearer insights into subsurface structures, tissue composition, and fluid dynamics, with the flexibility to seamlessly transition back to the original coordinate system post-interpretation. This interdisciplinary synergy underscores the profound impact of fundamental theoretical research on advancing practical applications across diverse scientific domains.

The development of a Digital Track Model (DTM) within Kazakhstan's railway system presents an innovative approach to enhancing efficiency and safety across the network. This paper investigates the economic viability of DTM implementation, offering a comprehensive mathematical model that highlights potential reductions in maintenance costs and improvements in system accuracy and durability. The findings advocate for DTM's potential to significantly elevate railway operations, suggesting a pivotal step towards modernizing Kazakhstan's transit infrastructure.

This manuscript presents a novel approach to estimating the potential within Schrödinger equations, with a particular focus on applications to the Navier-Stokes problem in fluid dynamics. By establishing new theoretical estimates, we delve into the intricate dynamics of fluid flow, aiming to unveil previously obscured aspects of the Navier-Stokes equations. Through a rigorous mathematical framework, we explore how these new potential estimates can provide a fresh perspective on fluid mechanics, contributing to the ongoing quest to solve some of its most persistent challenges.

This article presents a description of a new method for solving the factorization problem, based on the gradient descent algorithm. This approach demonstrates significant improvements in efficiency compared to traditional methods proposed in previous research. This article describes a new method for solving the factorization problem, based on the gradient descent algorithm, and demonstrates the transition from algebraic methods to approaches based on functional analysis. The proposed approach not only improves the efficiency of solving the problem but also allows the application of computational algorithms of functional analysis, opening new possibilities for research and optimization.