IJPAM: Volume 60, No. 4 (2010)
HYPOTHESIS USING A VARIANT OF
THE DIRICHLET ETA FUNCTION



Wildwood School
11811, Olympic Blvd., Los Angeles, California, 90064, USA


Abstract.The Riemann zeta function is
and the hypothesis is equivalent to the statement that all zeros of the Dirichlet eta function
have a
. For the purpose of this study,
will now be a complex number and
, where the independent variable becomes a selected set of
values. Using the trigonometric form of the denominator,
, the eigenvalues and eigenvectors are
and
, respectively. Since the eigenvalues are complex,
is rotated from
and we can set
and
to be constant, chosen from
. This approach first demonstrates the viability of the critical strip,
, and then solving for a particular eigenvector yields
. Each value of
determines values of
and
from the equations derived. In this plausibility argument for the Riemann Hypothesis, a process for selecting eigenvectors results in reducing the real part of the eta function to as close to zero as is arbitrarily desired. Additionally, the methods and results of this study are applied to the question of the Mass Gap Hypothesis in Appendix.
Received: March 17, 2010
AMS Subject Classification: 11M41, 11C20, 81S99
Key Words and Phrases: Riemann Hypothesis, Dirichlet eta function, eigenvectors, convergence to zero, Mass Gap Hypothesis
Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 60
Issue: 4