Partial sums of tan n · affine·mess

affine·mess f(N) = Σ_k≤N tan k N –

The partial sums of tan n

The graph of f(N) = tan 1 + tan 2 + ⋯ + tan N for 1 ≤ N ≤ 10^8, or up to 10^10 after Extend to 10¹⁰ (asking for any range past 10^8 extends automatically). Drag to pan; scroll or pinch to zoom; double-click to zoom in (shift-double-click to zoom out); shift-drag across a stretch of N to zoom into it. You can also type the range: entries like 1e6, 10^7 or 2,500,000 work. The vertical scale always refits to what is on screen.

A first heuristic

Since π is irrational, n mod π is equidistributed, so the values tan n are distributed like the tangent of a uniformly random angle: a standard Cauchy distribution. A sum of N independent Cauchy variables is again Cauchy, scaled by N: no averaging out, no √N. Stretch the horizontal axis by λ and the typical vertical size grows by λ as well, which is why an auto-scaled window looks alike at every scale. The terms are not independent, though, and the precise picture (below) comes from their correlations.

Where the cliffs come from

|tan n| is large when n is close to an odd multiple of π/2, that is, when 2n/(2m+1) is an unusually good approximation of π. The record values of |tan n| come from convergents and semiconvergents of π:

n	tan n	2n/(2m+1)
11	−225.95	22 / 7
52,174	−181,570	104,348 / 33,215
260,515	383,611	521,030 / 165,849
573,204	−3,402,634	1,146,408 / 364,913
42,781,604	−85,369,290	85,563,208 / 27,235,615
122,925,461	326,900,724	245,850,922 / 78,256,779
534,483,448	1,914,547,469	1,068,966,896 / 340,262,731
3,083,975,227	13,356,993,780	6,167,950,454 / 1,963,319,607

Rows past 10^8 extend the range when clicked.

Staircases

Because 355 ≈ 113π (off by 3.0·10^−5), tan(n+355) is almost tan n, with the angle nudged slightly toward the pole. Starting from n = 344 the terms at 344, 699, 1054, … all sit just below a pole, with values 227.5, 229.1, 230.7, … creeping upward, and the partial sums climb a long staircase. After 146 steps, at n = 52,174, the angle crosses the pole, the tangent flips to −181,570, and the climb is undone in a single term. See it on [1, 60,000]. Zoom far in anywhere and the near-periods 22 ≈ 7π and 355 ≈ 113π show up as approximate repetitions: [1, 400].

Self-similarity, made precise

Since tan x = −cot(x + π/2), the function f belongs to the two-parameter family of cotangent sums over a rotation:

S_{α,β}(N) = Σ_{n≤N} cot π(nα+β), f = −S_{1/π,1/2}.

The key identity is that for odd q

Σ_{j<q} tan(x + jπ/q) = q tan qx.

Take q = 355 and ε = 355 − 113π ≈ 3.0144·10^−5. Over any 355 consecutive integers, n/π runs through the residues j/355 up to a slow drift, so every block of 355 terms sums to

f(m+355) − f(m) ≈ 355 tan(ε(m+178)),

with relative error about 1/292 (292 is the partial quotient of π that follows 355/113). On scales beyond 355, then, f is a Riemann sum of a tangent with a tiny step. Summing it, and treating the terms next to each pole separately with Σ 1/(n+θ) = π cot πθ, gives

f(N) ≈ C − (1/ε) log|2 sin ε(N−N0)| + (π/ε) Σ_{N_k<N} cot π(θ0 + kρ),

where the poles are at N_k = N_0 + kπ/ε with N_0 ≈ 52,109.1 and period π/ε ≈ 104,218.28, θ_0 ≈ 0.8173, and ρ = G⁵(1/π) = 0.634591… is the fifth iterate of the Gauss map G(x) = {1/x}. The logarithm draws the arcs that rise into each cusp; the cotangent term is the jump at each pole. Numerically the fit is very good: at all 959 poles below 10^8 the jumps match (π/ε) cot πθ_k to within 133 on average (a typical jump is about 485,000), the phases θ_k advance by exactly ρ, and what remains of f is constant to within a standard deviation of 280.

The jumps themselves form S_{ρ,θ0}, a sum from the same family. So the graph on [1, 10^8] is the graph of S_{ρ,θ0} on [1, 960], magnified by λ = π/ε ≈ 10^5 in both directions, with logarithmic arcs attached. Zooming out by λ maps the family to itself and moves the rotation number along the Gauss map, 1/π ↦ G⁵(1/π): the partial quotients 3, 7, 15, 1, 292 have been used up, and the 292 went into the smooth arcs.

The next level repeats the pattern. ρ = [0;1,1,1,2,1,3,1,14,…] has the convergent 33/52 followed by the partial quotient 14, so every 52nd pole has θ creeping toward 0. That gives a staircase of cliffs at 4,846,147, 10,265,498, …, 37,362,253, undone at 42,781,604, where θ ≈ 0.9996. Here 52 × 104,218 ≈ 5,419,351 ≈ 1,725,033π plays the role of 355, and the 14 in π = [3;7,15,1,292,1,1,1,2,1,3,1,14,…] plays the role of 292. See [1, 10⁸]. Length and height scale by the same factor at every level, which is the precise content of the heuristic above.

When is it exactly self-similar?

Exact self-similarity needs the renormalization to come back to where it started, that is, a periodic continued fraction: a quadratic irrational. For the golden rotation α = (√5−1)/2, Knill proved that the rescaled cotangent sums converge, along Fibonacci scales, to a limit function with s(αx) = −α s(x), a genuinely self-similar graph. The same happens for g(N) = Σ tan(πnα): the curves t ↦ g(⌊tF_{3k}⌋)/F_{3k} converge to (−1)^k H(t), with differences shrinking by φ³ ≈ 4.24 per step, and therefore H(α³t) = −α³H(t). Since π is transcendental its continued fraction never repeats, and the same comparison at the scales 355, 103,993, 104,348, … gives differences of order 1 that never settle. There is no single limiting shape, only a sequence of renormalized siblings.

Is the graph a fractal?

Not in the sense of dimension. Box counting on [1, 10^8] gives local slopes of 1.04 to 1.06, the signature of a count growing like δ^{−1} log(1/δ), and a random walk with Cauchy steps behaves the same way. This matches the heuristic: by Blumenthal and Getoor the graph of an α-stable process has Hausdorff dimension max(1, 2 − 1/α), which is 1 for the Cauchy case α = 1. What the graph has instead is structure at infinitely many levels, the same scaling in both directions, and a total variation that diverges only logarithmically.

What is proved

Each renormalization step can be checked as above; it is clean when the next partial quotient is large and only approximate when it is small. For almost every α, with a random phase, Sinai and Ulcigrai proved that S_N/N has a limiting distribution: a statistical form of self-similarity, driven by the ergodicity of the Gauss map. For π itself such statements would need its continued fraction to be typical, which is wide open; it is not even known whether the partial quotients of π are bounded.

Beyond 10⁸

Between 10^8 and 10^9 two convergents of π dominate, and this time both terms are positive: tan 122,925,461 ≈ 3.27·10^8 and tan 534,483,448 ≈ 1.91·10^9, lifting the curve to f(10^9) ≈ 1.626·10^9. See [1, 10⁹].

The next convergent, 6,167,950,454 / 1,963,319,607, produces the largest term below 10^10: tan 3,083,975,227 ≈ 1.336·10^10, more than four times its own index. A single step lifts f from about 1.5·10^9 to 1.48·10^10, and the rest of the range, with cliffs at 5,633,467,006 (−2.68·10^9) and 9,251,925,681 (4.45·10^9), ends at f(10^10) ≈ 1.716·10^10. See [1, 10¹⁰].

How big can tan n be?

How it is computed

Every term is computed once, in background threads, with compensated (Neumaier) summation in double precision; JavaScript's Math.tan reduces integer arguments modulo π accurately. Up to 10^8 the applet stores f at every 256th N together with the minimum and maximum of f on each block (about 6 MB); beyond 10^8 it uses blocks of 4,096 (about 39 MB up to 10^10), plus an overview level of 65,536. Windows up to 400,000 wide are recomputed term by term from the nearest checkpoint, so every plotted point is exact; wider windows are drawn from the block extremes, which still show every spike, and narrow windows past 10^8 are refined by a background worker a moment after you stop. An independent extended-precision computation agrees to about 15 significant digits: f(10^8) = −55,500,322.5230…, f(10^9) = 1,625,640,367.0370… and f(10^10) ≈ 17,160,408,646.91. On a typical laptop 10^8 takes a few seconds and 10^10 a few minutes; anything already computed can be explored right away.

Hover (or tap) to read off N, f(N) and tan N. Keys: ← → pan, + − zoom. The URL keeps the current range, so a view can be shared.