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Logarithmic law for the proportion of sinks among critical-line zeros

Determine whether there exist constants a,b ∈ ℝ and a sufficiently large cutoff n0 such that, for n ≤ n0, the proportion P_n of sinks among the first n nontrivial zeros on the critical line satisfies P_n ∼ a log(n) + b.

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Background

The authors examine numerical data on the classification of nontrivial zeros on the critical line into sinks and sources for the holomorphic flow s'=ζ(s), and introduce P_n as the fraction of sinks among the first n zeros.

Motivated by empirical observations (including plots in the paper), they conjecture a specific logarithmic trend for P_n with respect to n, suggesting a slowly varying growth rate in the prevalence of sinks relative to all zeros.

References

We conjecture that the proportion of sinks vs. nontrivial zeros follows a logarithmic distribution of the form $P_n\sim a\log(n)+b $, for some $a,b\in R$ and $n \leq n_0$ fixed sufficiently large.

— The Generalized Riemann Zeta heat flow (2402.10154 - Castillo et al., 15 Feb 2024) in Remark (On stable and unstable critical points of the holomorphic flow), Section 1 (Introduction and Main Results)