Dear David, I want to state first some prelimnary remarks. Do you know about the universality of the Riemann Zeta function? The behaviour in the region $1/2 < \Re s < 1$ of the Riemann zeta function is chaotic. In fact, given $\epsilon >0$, for any compact region $D$ in $1/2 < \Re s < 1$ ans any non vanishing bounded holomorphic function $f$ on $D$, there exists a sequence $T_n = \Omega(n)$ such that $$ \sup_{z \in D} | f(z) -\zeta(z + iT_n)| < \epsilon,$$ or even stronger the measure of all such $T$ has lower positive density. Here are the precise statements http://en.wikipedia.org/wiki/Zeta_function_universality.

Perhaps a related fact: The Riemann hypothesis holds if and only if $f$ can be replaced here by the Riemann Zeta function! Proof for $<=$: Assume that $\zeta$ can be replaced for $f$ anywhere and $RH$ fails once say for a point in some $D$, then $\zeta$ would approximate itself on this $D$ arbitrary good in linear time. This produced every time a zero by Rouche's principle, which are far to many zeros by contradicting density results for zeros. Poof For $=>$: If $\zeta$ fulfils RH, we can replace $f$ by $\zeta$.

So since the rescaling factor for $Z$ is pretty regular, hence you can deduce this quasiperiodic property in the region $1/2 < Re s <1$ directly from the property known for the Riemann Zeta function. (Approximate a continous function by an entire via the theorem of Mergelyan). On the critical line Joern Steuding & Co. have presented some results last year also for $Re s = 1/2$, which imply conjecture $A$. In fact, I think he mentioned that it becomes more and more difficult if you replace $\zeta$ by its derivatives, which is merely a corollary in the region $1/2 < Re s < 1$.

Dear David, I want to state first some prelimnary remarks. Do you know about the universality of the Riemann Zeta function? The behaviour in the region $1/2 < \Re s < 1$ of the Riemann zeta function is chaotic. In fact, given $\epsilon >0$, for any compact region $D$ in $1/2 < \Re s < 1$ ans any non vanishing bounded holomorphic function $f$ on $D$, there exists a sequence $T_n = \Omega(n)$ such that $$ \sup_{z \in D} | f(z) -\zeta(z + iT_n)| < \epsilon,$$ or even stronger the measure of all such $T$ has lower positive density. Here are the precise statements http://en.wikipedia.org/wiki/Zeta_function_universality.

Perhaps a related fact: The Riemann hypothesis holds if and only if $f$ can be replaced here by the Riemann Zeta function! Proof for $<=$: Assume that $\zeta$ can be replaced for $f$ anywhere and $RH$ fails once say for a point in some $D$, then $\zeta$ would approximate itself on this $D$ arbitrary good in linear time. This produced every time a zero by Rouche's principle, which are far to many zeros by contradicting density results for zeros. Poof For $=>$: If $\zeta$ fulfils RH, we can replace $f$ by $\zeta$.

So since the rescaling factor for $Z$ is pretty regular, hence you can deduce this quasiperiodic property in the region $1/2 < Re s <1$ directly from the property known for the Riemann Zeta function. (Approximate a continous function by an entire via the theorem of Mergelyan). On the critical line Joern Steuding & Co. have presented some results last year also for $Re s = 1/2$, which probably imply conjecture $A$.

Dear david, I want to state first some prelimnary remarks. Do you know about the universality of the Riemann Zeta function? The behaviour in the region $1/2 < \Re s < 1$ of the Riemann zeta function is chaotic. In fact, given $\epsilon >0$, for any compact region $D$ in $1/2 < \Re s < 1$ ans any non vanishing bounded holomorphic function $f$ on $D$, there exists a sequence $T_n = \Omega(n)$ such that $$ \sup_{z \in D} | f(z) -\zeta(z + iT_n)| < \epsilon,$$ or even stronger the measure of all such $T$ has lower positive density.

Perhaps a related fact: The Riemann hypothesis holds if and only if $f$ can be replaced here by the Riemann Zeta function! Proof for $<=$: Assume that $\zeta$ can be replaced for $f$ anywhere and $RH$ fails once say for a point in some $D$, then $\zeta$ would approximate itself on this $D$ arbitrary good in linear time. This produced every time a zero by Rouche's principle, which are far to many zeros by contradicting density results for zeros. Poof For $=>$: If $\zeta$ fulfils RH, we can replace $f$ by $\zeta$.

So since the rescaling factor for $Z$ is pretty regular, hence you can deduce this quasiperiodic property in the region $1/2 < Re s <1$ directly from the property known for the Riemann Zeta function. (Approximate a continous function by an entire via the theorem of Mergelyan). On the critical line Joern Steuding & Co. have presented some results last year also for $Re s = 1/2$, hence that should be also doable, hence conjecture $A$ should be true from this perspective.

Dear David, I want to state first some prelimnary remarks. Do you know about the universality of the Riemann Zeta function? The behaviour in the region $1/2 < \Re s < 1$ of the Riemann zeta function is chaotic. In fact, given $\epsilon >0$, for any compact region $D$ in $1/2 < \Re s < 1$ ans any non vanishing bounded holomorphic function $f$ on $D$, there exists a sequence $T_n = \Omega(n)$ such that $$ \sup_{z \in D} | f(z) -\zeta(z + iT_n)| < \epsilon,$$ or even stronger the measure of all such $T$ has lower positive density. Here are the precise statements http://en.wikipedia.org/wiki/Zeta_function_universality.

Perhaps a related fact: The Riemann hypothesis holds if and only if $f$ can be replaced here by the Riemann Zeta function! Proof for $<=$: Assume that $\zeta$ can be replaced for $f$ anywhere and $RH$ fails once say for a point in some $D$, then $\zeta$ would approximate itself on this $D$ arbitrary good in linear time. This produced every time a zero by Rouche's principle, which are far to many zeros by contradicting density results for zeros. Poof For $=>$: If $\zeta$ fulfils RH, we can replace $f$ by $\zeta$.

So since the rescaling factor for $Z$ is pretty regular, hence you can deduce this quasiperiodic property in the region $1/2 < Re s <1$ directly from the property known for the Riemann Zeta function. (Approximate a continous function by an entire via the theorem of Mergelyan). On the critical line Joern Steuding & Co. have presented some results last year also for $Re s = 1/2$, which imply conjecture $A$. In fact, I think he mentioned that it becomes more and more difficult if you replace $\zeta$ by its derivatives, which is merely a corollary in the region $1/2 < Re s < 1$.