Higher Derivatives of the Hurwitz Zeta Function

<p>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).</p>

Name thesis:degree_name

Master of Science

Discipline thesis:degree_discipline

Department of Mathematics and Computer Science

Year

2011

Author dc:creator

• Musser, Jason

Contributors dc:contributor

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

• Riemann Zeta Function
• Stieltjes Constants
• Series expansion
• Functional equation
• Mathematics
• Number Theory

Repository record dc:identifier

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

OAI identifier oai:identifier

oai:digitalcommons.wku.edu:theses-2093