#### Publication Date

8-2011

#### Advisor(s) - Committee Chair

Dr. Dominic Lanphier (Director), Dr. Tilak Bhattacharya, Dr. Claus Ernst

#### Degree Program

Department of Mathematics and Computer Science

#### Degree Type

Master of Science

#### Abstract

The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s,q) with respect to s at s = 0. As a consequence, we obtain relations among certain important constants, the generalized Stieltjes constants. We also give numerical estimates of several values of the derivatives of ζ(s,q).

#### Disciplines

Mathematics | Number Theory

#### Recommended Citation

Musser, Jason, "Higher Derivatives of the Hurwitz Zeta Function" (2011). *Masters Theses & Specialist Projects.* Paper 1093.

http://digitalcommons.wku.edu/theses/1093

*Addendum to Higher Derivatives of the Hurwitz Zeta Function*

## Comments

Dominic Lanphier, in the Addendum, reviews some results of Stieltjes constants and relates them to the thesis. Mark Coffey is thanked for bringing attention to this work.