Abstract
This paper studies zeta functions of the form \(\sum\nolimits_{n = 1}^\infty {\chi (n){n^{- s}}}\), with χ a completely multiplicative function taking only unimodular values. We denote by σ(χ) the infimum of those α such that the Dirichlet series \(\sum\nolimits_{n = 1}^\infty {\chi (n){n^{- s}}}\) can be continued meromorphically to the half-plane Re s > α, and denote by ζχ(s) the corresponding meromorphic function in Re s > σ(χ). We construct ζχ(s) that have σ(χ) ≤ 1/2 and are universal for zero-free analytic functions on the half-critical strip 1/2 < Re s < 1, with zeros and poles at any discrete multisets lying in a strip to the right of Re s = 1/2 and satisfying a density condition that is somewhat stricter than the density hypothesis for the zeros of the Riemann zeta function. On a conceivable version of Cramér’s conjecture for gaps between primes, the density condition can be relaxed, and zeros and poles can also be placed at β + iγ with β ≤ 1 − λ log log ∣γ∣/ log∣γ∣ when λ > 1. Finally, we show that there exists ζχ(s) with σ(χ) ≤ 1/2 and zeros at any discrete multiset in the strip 1/2 < Re s ≤ 39/40 with no accumulation point in Re s > 1/2; on the Riemann hypothesis, this strip may be replaced by the half-critical strip 1/2 < Re s < 1.
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Acknowledgements
I am indebted to Éric Saïas for introducing me to Bohr’s approach to the Riemann hypothesis, for many inspiring discussions, and for valuable and constructive feedback on preliminary versions of this manuscript. I would also like to thank Andriy Bondarenko, Danylo Radchenko, Éero Saksman, Christian Skau, and Michel Weber for some helpful comments. Finally, I am indebted to the anonymous referee for a thorough review that helped me remove several inaccuracies from the text.
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Dedicated to Lawrence Zalcman with admiration
Research supported in part by Grant 275113 of the Research Council of Norway.
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Seip, K. Universality and distribution of zeros and poles of some zeta functions. JAMA 141, 331–381 (2020). https://doi.org/10.1007/s11854-020-0126-3
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DOI: https://doi.org/10.1007/s11854-020-0126-3