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The curve t↦ζ(1/2+it)t\mapsto\zeta(1/2+it) traced by the Riemann zeta-function along the critical line is the central object of the Riemann Hypothesis, yet its elementary geometry is largely uncharted. Two linked questions, highlighted by Sourmelidis and Steuding (2023) in their study of zeta spirals, remain open: (1) is the value set {ζ(1/2+it):t∈R}\{\zeta(1/2+it):t\in\mathbb{R}\} dense in the complex plane? (Bohr–Courant theory gives denseness for every fixed line 1/2<σ≤11/2<\sigma\le 1, and universality provably breaks down at σ=1/2\sigma=1/2, leaving this folklore conjecture — often attributed to Ramachandra — untouched); and (2) does the curvature of the critical-line spiral have a fixed sign for large tt unconditionally, as Gonek and Montgomery proved only under the Riemann Hypothesis? Both questions are open as stated, with the denseness question as the load-bearing core.

Background

The Riemann zeta-function ζ(s)\zeta(s), defined for Re s>1\mathrm{Re}\,s>1 by ζ(s)=∑n≥1n−s\zeta(s)=\sum_{n\ge1}n^{-s} and extended analytically to the whole complex plane except a simple pole at s=1s=1, is the central special function of analytic number theory. Its nontrivial zeros are conjectured to lie on the critical line Re s=1/2\mathrm{Re}\,s=1/2 (the Riemann Hypothesis, RH). This problem concerns the plane curve — the zeta spiral — traced by the values ζ(σ+it)\zeta(\sigma+it) as the ordinate tt varies, with σ\sigma fixed.

For 1/2<σ≤11/2<\sigma\le 1, Harald Bohr and Richard Courant proved in 1914 that the value set ζ(σ+iR)\zeta(\sigma+i\mathbb{R}) is dense in C\mathbb{C}. This is also a consequence of Voronin's universality theorem (1975): on every compact subset KK of the strip 1/2<Re s<11/2<\mathrm{Re}\,s<1 with connected complement, any continuous non-vanishing target function that is analytic in the interior can be uniformly approximated by vertical shifts ζ(s+iτ)\zeta(s+i\tau) for a set of shifts τ\tau of positive lower density; choosing a constant target gives denseness. Universality, however, cannot hold on the critical line itself: the shifted functions ζ(s+iτ)\zeta(s+i\tau) inherit the abundant zeros of ζ\zeta near Re s=1/2\mathrm{Re}\,s=1/2, which obstructs approximating zero-free targets there. Indeed, the region to the left encodes RH exactly: Sourmelidis and Steuding record (their Theorem 5) that RH holds if and only if ζ(σ+iR)\zeta(\sigma+i\mathbb{R}) is nowhere dense in C\mathbb{C} for every fixed σ<1/2\sigma<1/2, strengthening earlier work of Garunkštis–Steuding and Lapidus cited in their paper. The line σ=1/2\sigma=1/2 is thus the sharp, untouched boundary for denseness, and it is a folklore conjecture — attributed by Thomas Christ to Ramachandra — that ζ(1/2+iR)\zeta(1/2+i\mathbb{R}) is dense in C\mathbb{C}. The best partial result, due to Christ (2014), is that there exist curves approaching the critical line asymptotically on which the zeta-values are dense in C\mathbb{C}.

The second thread is local differential geometry of the same curve. The curvature of the parametrized curve t↦ζ(σ+it)t\mapsto\zeta(\sigma+it) (the rate at which its tangent direction turns, computed as κ(t)=Re{ζ′′ζ′‾/∣ζ′∣3}\kappa(t)=\mathrm{Re}\{\zeta''\overline{\zeta'}/|\zeta'|^{3}\} wherever ζ′≠0\zeta'\ne 0) has the same sign as Re ζ′′/ζ′(σ+it)\mathrm{Re}\,\zeta''/\zeta'(\sigma+it), so curvature questions are questions about the logarithmic derivative of ζ′\zeta' and hence about the zeros of ζ′\zeta'. Steven Gonek and Hugh Montgomery showed, assuming RH, that the critical-line spiral turns clockwise — negative curvature — for all sufficiently large tt. Unconditionally, Sourmelidis and Steuding prove negative curvature for σ≤0\sigma\le 0 and for σ≥A\sigma\ge A with some A∈[3,4]A\in[3,4], and infinitely many curvature sign changes for σ∈(1/2,1+θ)\sigma\in(1/2,1+\theta); the sign at exactly σ=1/2\sigma=1/2 unconditionally is open and is RH-sensitive, since by a classical theorem of Speiser quoted in their proof, RH is equivalent to ζ′\zeta' having no zeros in the left half 0<Re s<1/20<\mathrm{Re}\,s<1/2 of the critical strip.

A related thread, secondary to the two questions above, concerns effectivity of universality at the strip boundary. A universality statement is called effective when the threshold T0T_0 up to which good shifts τ\tau are guaranteed to exist is computable from the data (target function, compact set, tolerance), rather than merely shown to exist. Effective uniform approximation is known for interior points σ0∈(1/2,1)\sigma_0\in(1/2,1) via Voronin's effective theorem on simultaneous approximation of derivatives, and Lamzouri, Lester and Radziwiłł (2018) proved an effective universality theorem for ζ\zeta giving an effective positive lower density for the set of good shifts; at the boundary point σ0=1\sigma_0=1, however, the approximation statement is known only non-effectively (existence of infinitely many good shifts, by a limit theorem of Voronin).

Problem Statement

The single overarching objective is an unconditional geometric theory of the zeta spiral t↦ζ(1/2+it)t\mapsto\zeta(1/2+it) on the critical line. As highlighted by Sourmelidis and Steuding (arXiv:2306.00460), two linked questions constitute this objective: (1) [global geometry — core] determine whether the value set {ζ(1/2+it):t∈R}\{\zeta(1/2+it):t\in\mathbb{R}\} is dense in C\mathbb{C} or not — the folklore conjecture attributed to Ramachandra asserts denseness; and (2) [local geometry] determine unconditionally the sign of the curvature of the curve t↦ζ(1/2+it)t\mapsto\zeta(1/2+it) for all sufficiently large tt — in particular, prove or disprove without recourse to the Riemann Hypothesis the clockwise turning (negative curvature, i.e. Re ζ′′/ζ′(1/2+it)<0\mathrm{Re}\,\zeta''/\zeta'(1/2+it)<0) established conditionally by Gonek and Montgomery. Results (1) and (2) describe the same critical-line curve globally and locally; the denseness question (1) is the core of this objective, and the unconditional curvature sign (2) is a linked open question of the same program.

References

  1. Athanasios Sourmelidis and Jörn Steuding, 'Spirals of Riemann's Zeta-Function — Curvature, Denseness and Universality', Math. Proc. Camb. Phil. Soc. 176 (2024) 325–338, DOI 10.1017/S0305004123000543 (preprint arXiv:2306.00460, DOI 10.48550/arXiv.2306.00460), https://arxiv.org/abs/2306.00460
  2. Steven M. Gonek and Hugh L. Montgomery, 'Spirals of the Zeta Function I', in: Analytic Number Theory (In Honour of Helmut Maier's 60th Birthday), Springer, 2015, pp. 127–131, DOI 10.1007/978-3-319-22240-0_9, https://doi.org/10.1007/978-3-319-22240-0_9
  3. Harald Bohr and Richard Courant, 'Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion', J. reine angew. Math. 144 (1914) 249–274, DOI 10.1515/crll.1914.144.249, https://doi.org/10.1515/crll.1914.144.249
  4. Thomas Christ, 'Value-distribution of the Riemann zeta-function and related functions near the critical line', preprint (2014), arXiv:1405.1553, https://arxiv.org/abs/1405.1553
  5. Justas Kalpokas, Maxim A. Korolev and Jörn Steuding, 'Negative values of the Riemann zeta function on the critical line', Mathematika 59 (2013) 443–462, DOI 10.1112/s0025579313000090, https://doi.org/10.1112/s0025579313000090
  6. Youness Lamzouri, Stephen Lester and Maksym Radziwiłł, 'An effective universality theorem for the Riemann zeta function', Comment. Math. Helv. 93 (2018) 709–736, DOI 10.4171/cmh/448 (preprint arXiv:1611.10325), https://doi.org/10.4171/cmh/448
  7. Saeree Wananiyakul, Jörn Steuding and Nithi Rungtanapirom, 'Effective Weak Universality in Short Intervals', preprint (2024), arXiv:2403.01787, https://arxiv.org/abs/2403.01787