On the arithmetic of the selberg class S

This thesis is a study of the Selberg class S, which plays an important role in analytic number theory. The class S consists of Dirichlet series that exhibit properties similar to those of the Riemann Zeta Function, on which a degree function is defined. S possesses a commutative monoid structure with a unit under the convolution product of Dirichlet series. A. Selberg made five conjectures based on the degree function defined on the monoid S. The accuracy of these predictions has profound implications, particularly in algebraic number theory. One of these conjectures is that the elements of S can be uniquely factorized through the primitive elements of S. Previous studies in this direction have typically extended the monoid S. In this thesis, we take a different approach by studying a special type of submonoid S0 of S and proving that S0 is a unique factorization monoid, thereby verifying Selberg's conjecture for this submonoid.