## Mahurin Honors College Capstone Experience/Thesis Projects

#### Department

Mathematics

#### Document Type

Thesis

#### Abstract

The Riemann Zeta Function is a function of vital importance in the study of number theory and other branches of mathematics. This is primarily due to its intrinsic link with the prime numbers of the ring of integers. The value of the Riemann Zeta Function at 0 and the values of the first few derivatives at 0 have been determined by various mathematicians. Apostol obtained a closed expression for the nth derivative of the Riemann Zeta Function at 0 that generalized previously known results. For higher derivatives, his result is useful for numerical computations. The Dedekind Zeta Function is a generalization of the Riemann Zeta Function which is used to study primes of more general rings of integers. In this paper we modify the methods of Apostol to study the derivatives of certain Dedekind Zeta Functions. In particular, we are interested in Dedekind Zeta Functions of complex quadratic extensions of the rational number field. We obtain a general, closed expression for the function and its first derivative evaluated at 0. We then provide a few explicit examples. The method allows us to obtain a closed expression for higher derivatives at 0 that is useful for explicit numerical computations.

#### Advisor(s) or Committee Chair

Professor Dominic Lanphier

#### Disciplines

Mathematics | Physical Sciences and Mathematics

#### Recommended Citation

Salazar, Nathan, "Derivatives of the Dedekind Zeta Function Attached to a Complex Quadratic Field Extention" (2010). *Mahurin Honors College Capstone Experience/Thesis Projects.* Paper 249.

https://digitalcommons.wku.edu/stu_hon_theses/249