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arXiv · 2207.08659

Helson zeta functions for characters with finitely many values

I. Bochkov
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Abstract

We show that the analytic continuations of Helson zeta functions $ ζ_χ(s)= \sum_1^{\infty}χ(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole critical strip $ 1/2 < \Re s <1 $ under Riemann Hypothesis for the function $ χ$ taking values in cubic roots of unity. If the sets are symmetric with respect to the real axis, the same can be achieved with $ χ$ taking values $ \pm 1 $.

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I. Bochkov. 2022-07-18. Helson zeta functions for characters with finitely many values. https://arxiv.org/abs/2207.08659

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