This work presents a comprehensive exploration of various aspects of number theory, divided into several key sections. The first section delves into Diophantine approximations and transcendental number theory, discussing polynomials with minimal deviation, Gelfond's theorem, and the arithmetic properties of E-functions. The second section focuses on forms and congruences, including discriminants of decomposable forms and solutions to linear congruences. The third section addresses metric number theory, employing trigonometric sums to estimate Hausdorff dimensions and examining the behavior of continued fractions. Multiplicative number theory is covered next, highlighting Selberg's inequality, smoothing inequalities, and the distribution of prime vectors in integral lattices. The fifth section investigates the distribution of additive functions, exploring the smoothness of arithmetic function distributions, the Erdős-Wintner theorem, and the law of large numbers concerning shifted primes. Finally, the work concludes with a discussion on zeta-functions and algebraic number theory, presenting formulae in analytic number theory and estimates for Hecke polynomials. The last section touches on computational number theory and its applications to numeric analysis, emphasizing the significance of normal bases in algebraic number fields. The contributions of various authors enrich the discourse throughout the text.
