Riemann Zeta Zeros — Critical Line Visualizer

Plots the Riemann–Siegel Z(t) function (real, zero-crossings on Re(s)=1/2 ≡ zeros of ζ(s)) and the trace of ζ(1/2+it) in the complex plane.

Scroll to zoom, drag to pan, double-click a chart to reset it.

Z(t) vs t — zero crossings = zeta zeros

ζ(1/2 + it) traced in the complex plane

Proportion of zeros proven on the critical line, by mathematician

Detected zeros (t where Z(t) = 0)

Methods, visually

Each proof pushes the bound with a different tool. The formulas below are the actual objects used — not illustrations of them.

1914 · Hardy

\(N_0(T) \to \infty \text{ as } T \to \infty\)

Studied Z(t) itself (the real function plotted above) and showed it changes sign infinitely often on t ∈ (0, T) — each sign change is a zero on the critical line. The chart below is literally Hardy's argument: crossings that never stop.

G. H. Hardy, Sur les zéros de la fonction ζ(s) de Riemann, C. R. Acad. Sci. Paris 158 (1914), 1012–1014. Predates arXiv (est. 1991) — no open-access link.

1942 · Selberg

\(N_0(T) \ge c \cdot N(T), \quad c > 0\)

First proof that a positive proportion (not just an infinite count) of zeros lie on the line, via mean-value estimates of ζ. The original proof gives no explicit numeric c — just that one exists.

c (unspecified, > 0)
A. Selberg, On the zeros of Riemann's zeta-function, Skr. Norske Vid. Akad. Oslo I. 10 (1942), 1–59. Predates arXiv — no open-access link.

1974 · Levinson

\(\displaystyle \psi(s) = \sum_{n \le y} \mu(n)\, n^{-s} \left(1 - \frac{\ln n}{\ln y}\right), \quad y = T^{1/2-\varepsilon}\)

The mollifier: a short Dirichlet polynomial that, multiplied against ζ, cancels its growth and straightens the spiral above — turning zero-detection into counting sign changes of a simpler function. This is the "mollifier length (y)" slider above.

N. Levinson, More than one third of zeros of Riemann's zeta-function are on σ=1/2, Adv. Math. 13 (1974), 383–436. Predates arXiv — no confirmed open-access link.

1989 · Conrey

\(\text{same } \psi\text{, longer \& higher-degree:} \quad y = T^{4/7-\varepsilon}\)

Same mollifier idea as Levinson, pushed further: a longer, higher degree Dirichlet polynomial combined with sharper 4th-moment estimates for ζ. More terms → tighter cancellation → more provable zeros.

Levinson: y-exponent 1/2
Conrey: y-exponent 4/7 ≈ 0.571
J. B. Conrey, At least two fifths of the zeros of the Riemann zeta function are on the critical line, Bull. AMS 20 (1989), 79–81 — ProjectEuclid. Full version: J. Reine Angew. Math. 399 (1989), 1–26.

2020 · Pratt, Robles, Zaharescu, Zeindler

\(\text{Levinson's method} + \text{twisted 2nd-moment} + \text{higher-degree mollifier} \Rightarrow 41.7\%\)

Layers a new zero-density input on top of the mollifier framework: zeros are provably sparser away from Re(s)=1/2, which is what lets a mollifier of bounded length catch a larger share of them. (Density sketch, not measured data.)

K. Pratt, N. Robles, A. Zaharescu, D. Zeindler, More than five-twelfths of the zeros of ζ are on the critical line, Res. Math. Sci. 7 (2020) — arXiv:1802.10521.

2026 · Claude — More Than Two Thirds…

\(\displaystyle \liminf_{T\to\infty} \frac{N_0^*(T,2T)}{N(T,2T)} \ge \frac{2}{3} \quad (\text{optimised: } 0.6725)\)

A different tool than the mollifier line above: Montgomery's 1973 pair-correlation sum, unconditionally. Off-line zero pairs {ρ, 1−ρ̄} contribute hyperbolic 2×2 blocks (signature (1,1)) to a Gram matrix; on-line simple zeros contribute clean positive-definite 1×1 blocks. Sylvester's law of inertia makes that split basis-independent, and a rank–trace inequality (proved via von Neumann's trace inequality — the matrix analogue of Montgomery's real-number step m² ≥ 2m−1) bounds how much of the matrix the off-line pairs can occupy. RHS is what used to need RH to read; here it's linear algebra instead.

Preprint, not yet peer-reviewed — the strongest claim in this README, but still a claim.

Claude, More Than Two Thirds of the Zeros of the Riemann Zeta Function Lie on the Critical Line (2026) — PDF. Not indexed on arXiv; hosted on Anthropic's CDN.