Browsing by Author "Pihlaja, Joonas"

In 1903 Landau published an article where he presented a proof of the Prime Number Theorem for natural numbers and the Prime Ideal Theorem for number fields, each concerning the distribution of prime objects in their respective structures. On examining his proof, it became apparent that the essential assumptions used by Landau were unique factorization and an asymptotic growth condition on the number of objects as their size increases. This inspired primarily Beurling, and later Bateman, Knopfmacher and others to extract and generalize the core components of Landau's proof to apply to more general number-like systems, that still retain the multiplicative properties of numbers and ideals. In the algebraic part of this thesis, we introduce the notion of an arithmetical semigroup that embodies the notion of unique factorization and measuring the size of an object. We study them mainly via the algebraic structure of functions from the arithmetical semigroups to the complex plane. Classical notions, such as Dirichlet convolution, summatory functions and the like, carry over in a straightforward manner. We next define additional structure that generalizes the notion of arithmetic progressions as equivalence classes of class groups of arithmetical semigroups. Character theory of finite abelian groups is then applied to the class groups to show necessary and sufficient conditions for equidistribution of objects in classes. In the analytical part of this thesis, we present the fundamentals of the theory of generalized Dirichlet series as a separate box of analytical tools, divorced from the preceding algebraic notions. We give proofs of basic results on domains of convergence, analytical continuation, and orders of zeros and poles. The main theorem of this thesis is a variation of the Wiener-Ikehara Theorem, proved using Newman's complex analytic method. The two parts are brought together in a proof of an Abstract Prime Number Theorem for Generalized Progressions. We then rederive classical results such as the quantitative version of Dirichlet's Theorem on arithmetic progressions, Prime Number Theorems for the natural numbers, generalized Beurling numbers satisfying Landau's growth condition, and for abelian groups. In the final part we discuss some historical paths in analytical number theory concerning the Prime Number Theorem and Dirichlet series.