The missing direction of the Soundararajan and Radziwiłł equivalence between small gaps of zeros of the Riemann zeta-function and the horizontal distribution of zeros of ζ′(s), and the conjectured small-v asymptotics of m(v) and m′(v)

Let ρ′ = β′ + iγ′ denote a zero of the derivative ζ′(s) of the Riemann zeta-function, and let γ⁺ be the ordinate following a zero ordinate γ of ζ(s). Soundararajan (1998) conjectured, on the Riemann Hypothesis (RH), that liminf (β′−1/2) log γ′ = 0 is equivalent to liminf (γ⁺−γ) log γ = 0; Zhang (2001) proved the (B)⇒(A) direction on RH, and the converse direction has resisted attack. Farmer and Ki (2012) introduced quantitative distribution functions m′(v) and m(v) for the normalized horizontal displacements of zeros of ζ′(s) and the normalized gaps between consecutive zeros of ζ(s), conjectured $m'(v)\sim\tfrac{8}{9\pi}v^{3/2}$ and $m(v)\sim\tfrac{\pi}{6}v^{3}$ as $v\to 0^{+}$, and showed that even weak forms of either asymptotic would rule out Landau–Siegel zeros. Radziwiłł (2014) conjectured the two-sided comparison $m(v/2\pi)\asymp m'(v^{2})$; Fan Ge (2017) proved one direction, $m(v/2\pi)\le m'(v^{2})$ for small $v$ on RH. The problem is to complete this program: prove the missing (A)⇒(B) direction and prove the reverse comparison so that m(v/2π) ≍ m′(v²) holds, or establish in their place the conjectured small-v asymptotics — the conjecturally sharp form of the same equivalence, which implies both. No resolution is known: the best later work (Stopple 2020, 2025) reduces a classified portion of the conjecture to an unproved curvature bound and provides numerical support, while the small-gap literature reaches m(v) > 0 only for explicit moderate v, not the v → 0 behavior.

Background

The Riemann zeta-function ζ(s) has its nontrivial zeros ρ = β + iγ in the critical strip 0 < β < 1; the Riemann Hypothesis (RH) asserts they all lie on the critical line β = 1/2. Write γ⁺ for the ordinate of the zero of ζ(1/2 + it) immediately following γ. A classical theorem of A. Speiser (1934) states that RH is equivalent to the derivative ζ′(s) having no zeros in the half-strip 0 < σ < 1/2, so the horizontal position of zeros ρ′ = β′ + iγ′ of ζ′(s) measures how far the zeros of ζ(s) are from critical-line behavior. This motivated N. Levinson and H. L. Montgomery's study of the zeros of ζ′(s) and K. Soundararajan's work on their horizontal distribution. Soundararajan (1998) conjectured that, assuming RH, the following two almost-certainly-true statements are equivalent: (A) $\liminf_{\gamma'\to\infty}(\beta'-\tfrac{1}{2})\log\gamma'=0$, i.e. zeros of ζ′(s) approach the critical line faster than 1/log γ′ along a subsequence; and (B) $\liminf_{\gamma\to\infty}(\gamma^{+}-\gamma)\log\gamma=0$, i.e. gaps between consecutive zeros of ζ(s) fall below any fixed multiple of the mean spacing (2π/log γ) along a subsequence. The content of the conjecture is a mechanism linking the two, independent of their individual truth. Y. Zhang (2001) proved on RH that (B) implies (A); the converse direction remains unproved, with the best partial results being due to M. Z. Garaev and C. Y. Yıldırım (2007) and to Fan Ge (2017).

To quantify the link, D. W. Farmer and H. Ki (2012) introduced distribution functions for the two statistics. With w the indicator function of the unit interval [0, 1], set

$$m'(v)=\liminf_{T\to\infty}\frac{2\pi}{T\log T}\sum_{T\le\gamma'\le 2T}w\!\left(\frac{(\beta'-1/2)\log T}{v}\right),\qquad m(v)=\liminf_{T\to\infty}\frac{2\pi}{T\log T}\sum_{T\le\gamma_n\le 2T}w\!\left(\frac{(\gamma_{n+1}-\gamma_n)\log T}{2\pi v}\right).$$

so m′(v) and m(v) measure the limiting lower density of zeros of ζ′(s) whose horizontal displacement and of consecutive zeta-gaps whose normalized length fall below v times the natural scale (log T for displacement, (log T)/2π for gaps). Random matrix theory supplies precise predictions: zeros of ζ(s) are believed to be distributed like eigenvalues of large random Hermitian matrices (the GUE model, underlying H. L. Montgomery's 1973 pair correlation conjecture, which predicts the two-point correlations of the ordinates γ), and F. Mezzadri's analysis (2003) of zeros of derivatives of characteristic polynomials leads to the conjecture $m'(v)\sim\tfrac{8}{9\pi}v^{3/2}$ as $v\to 0^{+}$, with the companion conjecture $m(v)\sim\tfrac{\pi}{6}v^{3}$ (Farmer–Ki). These are deep statements: J. B. Conrey and H. Iwaniec (2002) and Farmer–Ki showed that even weak forms of either asymptotic (e.g. m(v) > 0 for all v > 0) would imply the non-existence of Landau–Siegel zeros — the exceptional real zeros of Dirichlet L-functions near s = 1 whose possible existence is a central open problem tied to class numbers of imaginary quadratic fields.

M. Radziwiłł (2014) proved on RH the Farmer–Ki conjecture that $m'(v)\gg v^{\alpha}$ with $\alpha<2$ forces $m(v)>0$ for all $v>0$ (throughout, $\ll$ and $\gg$ denote Vinogradov's comparison symbols: $f\ll g$ means $|f|\le Cg$ for some constant $C>0$, and $f\gg g$ means $g\ll f$), and posed the finer conjecture that, assuming some polynomial lower bound $m(v)\gg v^{A}$ or $m'(v)\gg v^{A}$, one has the two-sided comparison m(v/2π) ≍ m′(v²) (≍ meaning each side is bounded above and below by positive constant multiples of the other). Fan Ge (2017) proved on RH, with no lower-bound hypothesis, the one-directional inequality m(v/2π) ≤ m′(v²) for all sufficiently small v, via a sharp geometric statement (Theorem 2 of that paper): if γ⁺ − γ < v/log γ for small v, then the box 1/2 < σ < 1/2 + v²/(4 log γ), γ ≤ t ≤ γ⁺ contains exactly one zero of ζ′(s). Ge's Theorem 3 gives a partial converse of an incomparable shape: on RH, liminf (β′ − 1/2)(log γ′)³ = 0 implies (B). Garaev–Yıldırım's earlier version, with the weaker hypothesis liminf (β′ − 1/2)(log γ′)(log log γ′)² = 0, has a right-hand side that would be trivially true if ζ(s) had infinitely many multiple zeros; Ge's result avoids that issue since γ⁺ > γ are ordinates of distinct zeros, at the cost of a stronger assumption on the left-hand side. Both are weaker than Soundararajan's conjecture. Thus, of the whole program, exactly one direction is in hand, and the converse direction, the reverse comparison, and the precise asymptotics all remain open.

Problem Statement

Assuming the Riemann Hypothesis (or, preferably, weaker or no hypotheses), complete the conjectured equivalence between the statistics of small gaps between consecutive zeros of ζ(s) and the horizontal distribution of zeros of ζ′(s). Specifically, with m′(v) and m(v) defined by Farmer and Ki as above:

(1) Prove that (A) implies (B) in Soundararajan's conjecture, i.e. that $\liminf_{\gamma'\to\infty}(\beta'-\tfrac{1}{2})\log\gamma'=0$ forces $\liminf_{\gamma\to\infty}(\gamma^{+}-\gamma)\log\gamma=0$, completing the equivalence with Zhang's (B)⇒(A) theorem.

(2) Prove the reverse comparison $m'(v^{2})\ll m(v/2\pi)$ for all sufficiently small $v>0$, under at most the mild polynomial lower-bound hypotheses ($m(v)\gg v^{A}$ or $m'(v)\gg v^{A}$) under which Radziwiłł posed it, so that, under the same hypotheses and together with Fan Ge's inequality m(v/2π) ≤ m′(v²), the full equivalence m(v/2π) ≍ m′(v²) holds.

(3) Establish the conjectured sharp form: $m'(v)\sim\frac{8}{9\pi}v^{3/2}$ and $m(v)\sim\frac{\pi}{6}v^{3}$ as $v\to 0^{+}$, which would imply both (1) and (2).

The asymptotics (3) are the conjecturally sharp form of the same objective and imply both (1) and (2). The predicted repulsion $\beta'-\tfrac{1}{2}\gg(\gamma')^{-2/3+o(1)}$ of zeros of ζ′(s) from the critical line and the removal of the RH assumption are natural strengthenings.

References

1. Fan Ge, "The distribution of zeros of ζ′(s) and gaps between zeros of ζ(s)", Advances in Mathematics 320 (2017), 574–594. DOI: 10.1016/j.aim.2017.09.009, https://doi.org/10.1016/j.aim.2017.09.009 (arXiv:1510.04359, https://arxiv.org/abs/1510.04359)
2. K. Soundararajan, "The horizontal distribution of zeros of ζ′(s)", Duke Mathematical Journal 91 (1998), 33–59. DOI: 10.1215/s0012-7094-98-09104-9, https://doi.org/10.1215/s0012-7094-98-09104-9
3. Yitang Zhang, "On the zeros of ζ′(s) near the critical line", Duke Mathematical Journal 110 (2001), 555–572. DOI: 10.1215/s0012-7094-01-11034-x, https://doi.org/10.1215/s0012-7094-01-11034-x
4. David W. Farmer and Haseo Ki, "Landau–Siegel zeros and zeros of the derivative of the Riemann zeta function", Advances in Mathematics 230 (2012), 2048–2164. DOI: 10.1016/j.aim.2012.04.020, https://doi.org/10.1016/j.aim.2012.04.020
5. Maksym Radziwiłł, "Gaps between zeros of ζ(s) and the distribution of zeros of ζ′(s)", Advances in Mathematics 257 (2014), 6–24. DOI: 10.1016/j.aim.2014.02.010, https://doi.org/10.1016/j.aim.2014.02.010 (arXiv:1301.3232, https://arxiv.org/abs/1301.3232)
6. Francesco Mezzadri, "Random matrix theory and the zeros of ζ′(s)", Journal of Physics A: Mathematical and General 36 (2003), 2945–2962. DOI: 10.1088/0305-4470/36/12/306, https://doi.org/10.1088/0305-4470/36/12/306
7. M. Z. Garaev and C. Y. Yıldırım, "On small distances between ordinates of zeros of ζ(s) and ζ′(s)", International Mathematics Research Notices 2007, Article rnm091. DOI: 10.1093/imrn/rnm091, https://doi.org/10.1093/imrn/rnm091
8. J. B. Conrey and H. Iwaniec, "Spacing of zeros of Hecke L-functions and the class number problem", Acta Arithmetica 103 (2002), 259–312. DOI: 10.4064/aa103-3-5, https://doi.org/10.4064/aa103-3-5
9. H. L. Montgomery, "The pair correlation of zeros of the zeta function", in Analytic Number Theory, Proc. Sympos. Pure Math. 24, American Mathematical Society (1973), 181–193. DOI: 10.1090/pspum/024/9944, https://doi.org/10.1090/pspum/024/9944 (zbMATH: https://zbmath.org/03422472)
10. Andreas Speiser, "Geometrisches zur Riemannschen Zetafunktion", Mathematische Annalen 110 (1934), 514–521. DOI: 10.1007/BF01448042, https://doi.org/10.1007/BF01448042 (zbMATH: https://zbmath.org/3015030)
11. Norman Levinson and Hugh L. Montgomery, "Zeros of the derivatives of the Riemann zeta-function", Acta Mathematica 133 (1974), 49–65. DOI: 10.1007/BF02392141, https://doi.org/10.1007/BF02392141 (zbMATH: https://zbmath.org/3450299)
12. Jeffrey Stopple, "Level Curves for Zhang's Eta Function", Experimental Mathematics (2025), 1–18. DOI: 10.1080/10586458.2025.2487172, https://doi.org/10.1080/10586458.2025.2487172 (arXiv:2503.07696, https://arxiv.org/abs/2503.07696)
13. Jeffrey Stopple, "X-Ray of Zhang's eta function", preprint (2020). arXiv:2009.11886, https://arxiv.org/abs/2009.11886
14. D. A. Goldston and C. L. Turnage-Butterbaugh, "A note on small gaps between zeros of the Riemann zeta-function", preprint (2019). arXiv:1904.06001, https://arxiv.org/abs/1904.06001
15. Hung M. Bui, Daniel A. Goldston, Micah B. Milinovich, Hugh L. Montgomery, "Small gaps and small spacings between zeta zeros", Acta Arithmetica 210 (2023). DOI: 10.4064/aa220731-15-2, https://doi.org/10.4064/aa220731-15-2
16. A. Simonič, T. Trudgian, C. L. Turnage-Butterbaugh, "Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function", preprint (2020). arXiv:2010.10675, https://arxiv.org/abs/2010.10675
17. Andrés Chirre, Felipe Gonçalves, David de Laat, "Pair Correlation Estimates for the Zeros of the Zeta Function via Semidefinite Programming", preprint (2018). arXiv:1810.08843, https://arxiv.org/abs/1810.08843
18. Fan Ge and Steven M. Gonek, "Critical Points of Polynomials With Roots on the Unit Circle", International Mathematics Research Notices 2024 (2024), no. 7, 5434–5457. DOI: 10.1093/imrn/rnad152, https://doi.org/10.1093/imrn/rnad152