Riemann Zeta Function Computed As Ζ(0. 5+yi+zi) : 3D Riemann Hypothesis?
Author: Jason Cole
Publisher:
Published: 2017-09-12
Total Pages: 67
ISBN-13: 9781549727511
DOWNLOAD EBOOK →In this book, I investigate (on a undergraduate level) a function similar to the Riemann zeta function, but with an additional free parameter: zeta(x+iy+iz), as a "3D" or "hyper-complex" zeta function. This researchstudies the analytic continuation of Zeta(1) and the zeros of this function, aiming to use this information to shed light on well-known problems in analytic number theory. Contained within this manuscript is a very short list of trivial zeros of zeta(x+yi+zi) to show data of how the trivial zeros form a discrete sawtooth Fourier wave in the 3D hypercomplex plane. The spectrum of a sawtooth wave mirrors the harmonic series. This research suggest that Zeta(1) is analytically continued into the rest of the complex plane as a discrete 3D sawtooth wave pattern made of trivial zeros. We can only observe the analytic continuation of Zeta(1) in the 3D hyper-complex plane as trivial zeros when Zeta is computed as s=x+yi+zi. This discovery of Zeta(1) analytically continued as the 3D trivial zeros (discrete sawtooth wave) suggest there is also a 3D hyper-complex landscape to the nontrivial zeros. A 3D or hyper-complex Riemann hypothesis were s=1/2+yi+zi. This book serves as the basis for 3D or hyper-complex analysis using computation were the Riemann Zeta function, similar L-functions, L-functions of Elliptic curves and Modular forms can be computed as s=x+yi+zi and give new insight into their properties that can't be seen when s=x+yi. This book also explore the topic of extending the Montgomery Pair correlation conjecture into the 3D or hyper-complex plane to correspond to the 3D or hyper-complex zeros of Riemann Zeta function.
