Browsing by Author "Legesse, Hunduma"
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Item Fractional Hypergeometric Zeta Functions(Addis Ababa University, 2014-05) Legesse, Hunduma; Hassen, Abdulkadir(Professor)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