### Ralph Furmaniak

Shared publicly -Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$: $f(z)$ is bounded when $\Re z>1+\delta$; $f(z)$ is unbounded when $\Re z=1$; $f(z)$ grows polynomiall...

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Great idea! You should post it.

You'd need bounds on the number of such zeros in intervals though. If you had a bounded number then you could get rid of these. Unfortunately I can't tell if you can do something given the Lindelof bound of o(log T); it's worth thinking about.

In the end you may just need to construct an artificial sequence of roots that are sufficiently regular at one scale but irregular at a slightly smaller scale.

You'd need bounds on the number of such zeros in intervals though. If you had a bounded number then you could get rid of these. Unfortunately I can't tell if you can do something given the Lindelof bound of o(log T); it's worth thinking about.

In the end you may just need to construct an artificial sequence of roots that are sufficiently regular at one scale but irregular at a slightly smaller scale.

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