Fractional Hypergeometric Zeta Functions
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Date
2014-05
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Addis Ababa University
Abstract
Since Riemann, there have been a great number of generalizations of the Riemann
zeta function. Of particular interest is the hypergeometric zeta functions a generalization
due to Hassen and Nguyen that satisfies many of the same properties as the
classical Riemann zeta function. In this dissertation we show that the second order
Hypergeometric Zeta Function has zero free region in the right half of the complex
plane and extend the hypergeometric zeta Functions to fractional hypergeometric zeta
functions. We rewrite the series representation of the second order hypergeometric
zeta function as a ”power series” with Dirichlet series as its coefficients. We use this
series representation to show that the second order hypergeometric zeta function has
zero free region to the right half of the complex plane. We generalize the hypergeometric
zeta function via the integral representation of the classical Riemann zeta
function to fractional hypergeometric zeta functions. It is shown that fractional hypergeomeric
zeta functions satisfy many of the properties satisfied by hypergeometric
zeta functions
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Fractional Hypergeometric Zeta Functions