IJPAM: Volume 60, No. 4 (2010)

A PLAUSIBILITY ARGUMENT FOR THE RIEMANN
HYPOTHESIS USING A VARIANT OF
THE DIRICHLET ETA FUNCTION

Alexander Sheftel$^1$, Lewis Forsheit$^2$
$^{1,2}$Department of Mathematics
Wildwood School
11811, Olympic Blvd., Los Angeles, California, 90064, USA
$^1$e-mail: alexanders10@wildwood.org
$^2$e-mail: lforsheit@wildwood.org


Abstract.The Riemann zeta function is $\zeta(s)=\sum\limits_{n=1}^\infty\frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$ and the hypothesis is equivalent to the statement that all zeros of the Dirichlet eta function $\eta(s)=\sum\limits_{k=1}^\infty\frac{(-1)^{(k-1)}}{k^s}=(1-2^{1-s})\zeta(s)$ have a $Re(s)=\frac{1}{2}$. For the purpose of this study, $k$ will now be a complex number and $\eta(\{b\})=\sum\limits_{a=1}^\infty(-1)^{(a+bi)-1}\frac{1}{(a+bi)^{(c+di)}}$, where the independent variable becomes a selected set of $b$ values. Using the trigonometric form of the denominator, $\left[ \begin{array}{c} \ln r_2\\ \theta_2\end{array}\right]=\left[ \begin{arra...
... \end{array} \right]\left[\begin{array}{c}-\ln r_1\\ \theta_1\end{array}\right]$, the eigenvalues and eigenvectors are $\lambda=c\pm di$ and $\left[ \begin{array}{c} \ln r_1\\ \mp i\ln r_1\end{array}\right]$, respectively. Since the eigenvalues are complex, $(r_2,\theta_2)$ is rotated from $(r_1,\theta_1)$ and we can set $r_2=r_1$ and $\theta_2$ to be constant, chosen from $\pm\frac{\pi}{3}, \pm\frac{5\pi}{3},\cdots$. This approach first demonstrates the viability of the critical strip, $0\le c\le1$, and then solving for a particular eigenvector yields $c=\frac{1}{2}$. Each value of $a$ determines values of $b$ and $d$ 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