Extended Zeta Functions Prove or Dis-prove Riemann's Hypothesis

Nonfiction, Science & Nature, Mathematics, Number Theory
Cover of the book Extended Zeta Functions Prove or Dis-prove Riemann's Hypothesis by James Constant, James Constant
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Author: James Constant ISBN: 9781311414205
Publisher: James Constant Publication: February 17, 2015
Imprint: Smashwords Edition Language: English
Author: James Constant
ISBN: 9781311414205
Publisher: James Constant
Publication: February 17, 2015
Imprint: Smashwords Edition
Language: English

While extended zeta functions support investigations of Riemann's hypothesis and estimates for the Prime Number Theorem, some zeta functions offer better prospects for providing easy proofs, or disproofs. In 1859, Riemann had the idea to define Euler’s function ε(x)=∑m^x for all complex numbers s=x+iy by analytic extension. This extension is important in number theory and plays a central role in the distribution of prime numbers. There are a number of ways of extending Euler's zeta function ζ(s) to points where 0≤x≤1. Because ζ(s) is an alternating series, it becomes possible to prove or disprove Riemann's Hypothesis.

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While extended zeta functions support investigations of Riemann's hypothesis and estimates for the Prime Number Theorem, some zeta functions offer better prospects for providing easy proofs, or disproofs. In 1859, Riemann had the idea to define Euler’s function ε(x)=∑m^x for all complex numbers s=x+iy by analytic extension. This extension is important in number theory and plays a central role in the distribution of prime numbers. There are a number of ways of extending Euler's zeta function ζ(s) to points where 0≤x≤1. Because ζ(s) is an alternating series, it becomes possible to prove or disprove Riemann's Hypothesis.

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