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Is there an analytic function with zeroes only at:

  • every $-2n$,
  • $\frac12\pm it$, and
  • at least one at $\frac12\pm\epsilon\pm it$ where $0<\epsilon<\frac12, t\neq0$ (and these zeroes observing the known reflective symmetries within the critical strip)?

A answer assuming the Riemann hypothesis true would be fine.

To my mind the uniqueness of any analytic continuation of $\zeta(s)$ is suggestive of the existence of such a function being incompatible with the Riemann Hypothesis.

If not, uniqueness of $\zeta$ is of course a nice simple sufficiency for the Riemann hypothesis.

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Take $\zeta(z)\left(z-\frac14-i\right)\left(z-\frac14+i\right)\left(z-\frac34-i\right)\left(z-\frac34+i\right)$.

Every condition that you mentioned holds.

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I think this is an open question. As long as your function is automorphic, the Grand Riemann Hypothesis asks this exact question.

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    $\begingroup$ Thanks, I didn't know that. Does the Grand Riemann Hypothesis ask if this is the case, and permit one to assume the Riemann hypothesis? Because if not, it would seem a potentially important result, if it were true, that to prove GRH assuming RH, would prove RH. $\endgroup$ – samerivertwice Sep 5 '18 at 9:39
  • $\begingroup$ That doesn't sound right. $\endgroup$ – Uri George Peterzil Sep 5 '18 at 15:34

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