The densities of the sets of natural numbers and applications in sequence spaces

The thesis consists of seven Chapters. In Introduction which is the Chapter one, we recall the general history of the main concepts, give some of the necessary information about our topic. In Chapter two, we mention the definition of natural density in some different literature and write some propositions, examples. Then we give the concepts of α-density, λ-density, 𝑓-density, and modulus function. Afterward, we give relationships between the natural density, α-density, λ-density, 𝑓-density. In Chapter three, we introduce some other density types, such as logarithmic, weighted, exponential, and Schnirelmann densities, and give the applications for each type. Later, we proceed with the comparisons of densities. In Chapter four, using densities, as applications of densities we define some kinds of convergence and boundedness for sequences, such as statistical convergence, statistical boundedness, and statistically Cauchy sequences. In Chapter five, we examine the concepts of λ-statistical convergence, λ-statistical boundedness. Moreover, give the inclusion relations between the sets S_λ and S_μ, [V,λ]^p and [V,μ]^p of number sequences. Furthermore, we introduce the concepts of 𝑓-statistical convergence, 𝑓-statistical boundedness, 𝑓-statistically Cauchy sequence, statistical convergence of order α, λ-statistical convergence of order α, 𝑓-statistical convergence of order α, statistical boundedness of order α, λ-statistical boundedness of order α. In Chapter six, we present the concept of Strong p-Cesàro summability of order α and additionally, we give the correlations between statistical convergence and strong p-Cesàro summability. The seventh chapter concludes with a thorough analysis of the thesis's results.