An analog computer running on Ramanujan’s 1914 series can produce digits of $1/\pi$ at rate $\approx 3120$ per unit time. Switching to Chudnovsky’s 1989 series — the one your Mathematica install actually uses — pushes this to $\approx 1.6 \times 10^{8}$. Can you go further? This post argues no, and the reason has nothing to do with analog computing: it is the Heegner–Stark–Baker theorem on imaginary quadratic fields with class number 1.

The setup

Consider an analog computer in the GPAC sense: a finite collection of state variables $(y_1, \dots, y_d)$ evolving according to a polynomial ODE

The computer “computes” a real number $\alpha$ if some component $y_1(\tau)$ converges to $\alpha$ as $\tau \to \infty$, and the rate of computation is the eigenvalue of the linearization at the fixed point: if $|y_1(\tau) - \alpha| \sim e^{-r \tau}$, then the analog computer produces $r/\log 10$ correct decimal digits per unit of time $\tau$.

For $\zeta(3) = \sum 1/n^3 \approx 1.202$, the best known analog construction reads $\zeta(3)$ out of the ratio of two formal-power- series solutions to a Picard–Fuchs ODE at its conifold singularity. This is the Apéry trick, exploited in an earlier post, and it produces an arbitrary-rate analog circuit for $\zeta(3)$.

For $\pi$, this conifold-ratio trick does not exist (the Ramanujan Picard–Fuchs operator has a one-dimensional formal-series kernel). What survives is a different mechanism — invert against the large quantity — which gives a finite, but bounded, speedup. The question of this post is: how bounded?

Ramanujan’s 9801

Start with Ramanujan’s 1914 identity:

Call the sum on the right $M_\infty \approx 1103.046\dots$. Then

The naive analog circuit for $\pi$ would compute $1/\pi$ to high precision (which is small, $\approx 0.318$) and invert it. Inverting a small number with the standard logistic gadget runs at rate $\approx 1/\pi$, which gives only $\approx 0.14$ digits per unit $\tau$ — slow.

The Ramanujan trick: compute $M_\infty$ instead, a large number, and invert that against the prefactor:

At the fixed point $P = 9801/(2\sqrt 2 \cdot M_\infty) = \pi$, the linearization eigenvalue is $-2 \sqrt 2 \cdot M_\infty \approx -3120$. So the analog computer produces $3120/\log 10 \approx 1355$ digits of $\pi$ per unit time $\tau$.

A factor of $9801 \approx 99^2$ over the naive rate, paid for once by the modular-form structure of Ramanujan’s identity.

Chudnovsky’s $640320^{3/2}$

In 1989, the Chudnovsky brothers found a faster series:

Each term contributes about $14.18$ decimal digits, more than Ramanujan’s $\approx 8$. By the same “invert against the large quantity” mechanism, the resulting analog circuit

runs at rate $12 M_\infty = 640320^{3/2}/\pi \approx 1.63 \times 10^{8}$ — about $52{,}000$ times faster than Ramanujan’s circuit.

So far this looks like a numerical race: maybe someone will find a $10^{12}$-faster series next year. But there is a structural reason to expect Chudnovsky to be the end of the line.

The j-invariant connection

Both Ramanujan’s $\frac{1}{\pi}$-series and Chudnovsky’s are instances of a general construction. Pick an imaginary quadratic field $K = \mathbb{Q}(\sqrt{-d})$ with $d$ squarefree. Pick a CM point $\tau \in K$ on the upper half-plane — a complex number whose ratio with its conjugate is in $K$. Klein’s modular function $j(\tau)$ takes special values at such CM points, and the resulting $j(\tau_d)$ is an algebraic integer of bounded degree.

Ramanujan-style $1/\pi$ identities are built from these CM values. The prefactor of the series at level $d$ scales (roughly) like $|j(\tau_d)|^{1/2}$ when $j(\tau_d)$ is rational, and the resulting analog rate is $|j(\tau_d)|^{1/2}/\pi$.

So: to get a faster $1/\pi$ analog circuit, find a CM point with a larger rational $j$-value.

Class number 1

Now Klein’s $j$-invariant takes a rational integer value at $\tau_d$ exactly when $K = \mathbb{Q}(\sqrt{-d})$ has class number 1. (Otherwise $j(\tau_d)$ is an algebraic integer over $\mathbb{Q}$ of degree equal to the class number, splitting over the Hilbert class field of $K$ — no clean rational identity.)

How many imaginary quadratic fields have class number 1? Carl Friedrich Gauss conjectured the answer should be finite. In 1934, Hans Heilbronn proved finiteness. In 1952, Kurt Heegner published a proof of the exact list — but the mathematical community did not accept it at the time, viewing the proof as incomplete. In 1966–1967, Harold Stark and Alan Baker gave independent, accepted proofs, which both Stark and Birch later confirmed had been correct in Heegner’s original argument all along.

The full list of squarefree $d > 0$ with $\mathbb{Q}(\sqrt{-d})$ of class number 1 is:

Nine values. The largest is $163$.

The corresponding $j$-values are (with $j(\tau_d) = J^3$, integer cubes for these special discriminants):

The $d = 163$ row gives Chudnovsky’s prefactor exactly. The next row down (d = 67) gives an analog rate already a factor of $10^3$ below Chudnovsky.

This is the structural ceiling: among class-number-1 CM points, $d = 163$ is the unique maximum, and there is no next one to discover.

Ramanujan’s near-integer

A consequence of $d = 163$ being a class-number-1 discriminant is that $\exp(\pi \sqrt{163})$ is almost an integer:

with $\varepsilon \approx 7.5 \times 10^{-13}$ (the next term in the $q$-expansion of $j$ is $196884 \cdot e^{-\pi \sqrt{163}}$, which is this small). Try it in your numerics: $\exp(\pi \sqrt{163}) = 262\,537\,412\,640\,768\,743.99999999999925\dots$

This near-integer relation is the same fact as Chudnovsky’s analog rate ceiling. The big number $640320$ is the cube root of $|j(\tau_{163})|$, and Chudnovsky’s series is essentially the Taylor expansion of a modular form at $\tau_{163}$, with the prefactor inheriting the magnitude of $|j|^{1/2}$.

Could you go higher?

Two ways to escape the Heegner ceiling:

(a) Higher class number. Imaginary quadratic fields with class number 2, 3, etc. exist (e.g., $\mathbb{Q}(\sqrt{-5})$ has class number 2). Their $j$-values are algebraic integers of degree equal to the class number — so $j(\tau_d)$ generates an extension. There are identities for $1/\pi$ involving such CM points, but they involve algebraic combinations and surds, and do not produce a single rational prefactor that one can plug into the analog inverter. Whether some recombination yields a useful continuous-time construction is open.

(b) Non-CM modular forms. Chudnovsky’s identity comes from a modular form on $\Gamma_0(163)$. There are higher-level modular constructions (with non-CM points), and there is the entire Calabi– Yau modular form taxonomy (Apéry-like sequences classified by Almkvist, van Straten, Zudilin). Most do not give $1/\pi$ identities at all; the AvSZ list of five $\pi$-producing CY families (hypergeometric level-1 quintic Calabi–Yau periods) all live within the class-number-1 CM regime and are bounded by Heegner-163.

So Chudnovsky-from-Heegner-163 appears to be a hard structural ceiling for the $\pi$-PIVP rate using known mechanisms.

Other paradigms?

What about completely different ways to compute $\pi$? A survey of the standard candidates:

• Machin-like arctan series: $\pi/4 = 4 \arctan(1/5) - \arctan(1/239)$ etc. The integrand is a rational function, so the PIVP encoding is straightforward; but the analog rate is $\log(\text{geometric ratio}) = O(1)$ digits per $\tau$. Slow. Why doesn’t the difference converge faster than either piece? See the discussion below.

• BBP (Bailey–Borwein–Plouffe digit-extraction): Same fate. Geometric series at base 16, analog rate $\log 16 \approx 2.77$ digits/$\tau$.

• Brent–Salamin AGM: Discrete iteration that doubles digits per step. The continuous-time PIVP encoding $\dot a = b - a$, $\dot b = (a-b) \cdot b/(2 b)$ runs at rate $\approx 1/2$ digits/$\tau$. Quadratic discrete convergence does not survive translation to a continuous-time ODE — the squaring step that doubles digits per iteration becomes a linear ODE with a single eigenvalue. This is a general phenomenon of discrete-vs-continuous computation: doubling is cheap discretely, but linear continuously.

• Borwein quartic iteration (1986): Same fate as AGM. Quartic discrete convergence becomes linear continuous.

• Brouncker continued fraction: $4/\pi = 1 + 1^2/(2 + 3^2/(2 + 5^2/\dots))$. Convergent rate: $O(1/k)$ per step, sublinear. Useless for analog.

So among all the named $\pi$-formulas in the textbook, Chudnovsky’s analog rate is the unique standout, and Heegner-163 explains why it is the maximum.

Why Machin-like arctan combinations don’t accelerate

A natural question: Machin’s identity

combines two arctan series, each of which converges geometrically. Could the combination converge faster than either piece, by some cancellation between truncation tails?

This is a real phenomenon in numerical analysis — sequence acceleration (Aitken $\Delta^2$, Shanks transformation, Euler acceleration) does exactly this kind of thing. So the question is not naive. Let us see why it fails for Machin specifically.

Machin’s identity is exact, not asymptotic: the equality $4 \arctan(1/5) - \arctan(1/239) = \pi/4$ holds for the full infinite series, so the truncation error of the combination is

where $R_N(x)$ is the tail of the arctan Taylor series at $x$ truncated after $N$ terms. The asymptotic form of the tail is

and the leading term depends on $x$. So the dominant terms of $R_N(1/5)$ and $R_N(1/239)$ are at different magnitudes, $(1/5)^{2N+1}$ vs $(1/239)^{2N+1}$, with the latter exponentially smaller. They cannot cancel: you have one big term and one small term, and their difference is dominated by the big one. Concretely

giving $\log_{10}(25) \approx 1.40$ digits per term. The $\arctan(1/239)$ piece, despite converging much faster on its own, contributes nothing to the rate.

For cancellation to help, the dominant tail terms would need to share algebraic structure — be scaled copies of the same sequence, modulo sign. The Taylor coefficients of $\arctan$ at $x = 1/5$ vs $x = 1/239$ do not share that structure. The constants $1/5$ and $1/239$ are algebraically independent rational points on the same Riemann surface (the projective line minus $\{i, -i\}$), but their Taylor expansions at zero are independent geometric series in $x^2$ with different ratios.

A stronger Machin-like family does exist: by choosing better integer arguments, one can push the rate up. For example,

gives $\log_{10}(64) \approx 1.81$ digits per term (rate-limited now by the slowest piece, $\arctan(1/8)$). Modern long Machin-like formulas (Hwang-Chien-Liu 1997, Wetherfield 2004) push the slowest $p$ in any $\arctan(1/p)$ up to $p \sim 10^5$, giving rates around 10 digits per term. But this is a linear improvement — the integers $p$ in valid Machin formulas are constrained by Gaussian-integer norm relations, with no known route to $p$ on the order of $10^8$.

The structural reason for this ceiling: $\arctan(z)$ satisfies the second-order linear ODE $(1+z^2) y'' + 2 z y' = 0$, with two singular points $z = \pm i$ and holonomic rank 2. Chudnovsky’s ${}_3 F_2$, by contrast, satisfies a third-order Picard–Fuchs operator with rank 3, and the underlying period structure is the $d=163$ CM modular surface, which carries an exponentially large algebraic invariant ($|j(\tau_{163})|^{1/2} = 640320^{3/2}$) as one of its periods. Holonomic rank 2 cannot produce algebraic invariants of magnitude $\sim 10^8$; rank 3 with CM does. This is a structural obstruction, not a clever-trick gap.

(For a careful PIVP construction of arctan-class circuits and a matching state-count comparison, see the next post.)

What’s next

The conjectural sharp ceiling for $\pi$-PIVP analog rate is $640320^{3/2}/\pi \approx 1.6 \times 10^{8}$ digits/$\tau$. Beating this would require either a class-number-$> 1$ CM construction that somehow produces a usable rational prefactor (currently mysterious) or a non-modular mechanism for analog computation of $\pi$ that does not yet exist in the literature.

For now, the Chudnovsky-via-Heegner-163 circuit appears to be the end of $\pi$.

Code: experiments/heegner_163_ceiling.py (j-invariant table, near-integer check, rate comparison), experiments/pi_non_avsz_survey.py (Machin, BBP, AGM, Borwein, Brouncker rates), experiments/chudnovsky_pi_pivp.py (the $640320^{3/2}/(12\pi)$ verification). Written as part of an overnight investigation, 2026-04-25.

「九层之台，起于累土。」

— 《道德经》

A nine-tiered tower rises from a basket of earth.