Note on a paper by H.\,L. Montgomery ``Omega theorems for the Riemann zeta-function''

K. Ramachandra and A. Sankaranarayanan

Abstract: We study Omega theorems for the expression $E=\RE(e^{i\th} \log \zeta(\s_0+it_0))$ where $1/2\le\s_0<1$ and $0\le\th<2\pi$ ($\s_0$, $\th$ fixed) as $t_0\to\bb$. In fact we prove $E\ge C(1-\s_0)^{-1}(\log t_0)^{1-\s_0} (\log\log t_0)^{-\s_0}$ for at least one $t_0$ in $[T^{\e},T]$ where $C$ is a positive constant. Note that $(1-\s_0)^{-1}\to\bb$ as $\s_0\to1$.

Keywords: Omega theorems, Riemann Zeta-function, Dirichlet box principle, Dedekind Zeta-function.

Classification (MSC2000): 10H05

Full text of the article:

• Compressed PostScript file (70 kilobytes)
• PDF file (181 kilobytes)