Extensions of Selberg-Delange method

Abstract

This dissertation involves two topics in analytic number theory. The first topic focuses on extensions of the Selberg-Delange Method, which are discussed in Chapters $2$ and $3$. The last topic, which is discussed in Chapter $4$, is a new identity for Multiple Zeta Values.

The Selberg-Delange method is a method that is widely used to determine the asymptotic behavior of the sum of arithmetic functions whose corresponding Dirichlet's series can be written in the term of the Riemann zeta function, $\zeta(s)$. In Chapter $2$, we first provide a history and recent developments of the Selberg-Delange method. Then, we provide a generalized version of the Selberg-Delange method which can be applied to a larger class of arithmetic functions. We devote Chapter $3$ to the proofs of the results stated in Chapter $2$.

In $1961$, Matsuoka evaluated $\zeta(2)$ by means of evaluating the integral $\ds \int_0^{\pi/2} x^{2}\cos^{2n}(x) dx$. The last chapter of this dissertation generalizes the idea of Matsuoka and obtains a new identity for Multiple Zeta Values.