Horizontal monotonicity of the modulus of the zeta function, $L$-functions, and related functions
Volume 166 / 2014
Abstract
As usual, let $s = \sigma + it$. For any fixed value of $t$ with $|t| \geq 8$ and for $\sigma < 0$, we show that $|\zeta(s)|$ is strictly decreasing in $\sigma$, with the same result also holding for the related functions $\xi$ of Riemann and $\eta$ of Euler. The following inequality related to the monotonicity of all three functions is proved: $$ \Re\biggl(\frac {\eta'(s)}{\eta(s)} \biggr) < \Re\biggl(\frac {\zeta'(s)}{\zeta(s)}\biggr) < \Re\biggl(\frac {\xi'(s)}{\xi(s)} \biggr). $$ It is also shown that extending the above monotonicity result for $|\zeta(s)|$, $|\xi(s)|,$ or $|\eta(s)| $ from $\sigma < 0$ to $\sigma < 1/2$ is equivalent to the Riemann hypothesis. Similar monotonicity results will be established for all Dirichlet $L$-functions $L(s,\chi)$, where $\chi$ is any primitive Dirichlet character, as well as the corresponding $\xi(s,\chi)$ functions, together with the relation of this to the generalized Riemann hypothesis. Finally, these results will be interpreted in terms of the degree $1$ elements of the Selberg class.
