Flint Hills Series: Partial Sums to $10^{10}$ with Spike Decomposition

Abstract

We computed the Flint Hills partial sum $S_{10^{10}} = \sum_{k=1}^{10^{10}} \frac{1}{k^3 \sin^2 k} = 30.31454612$ in 1.9 seconds on a single RTX 5090 GPU. This extends the computational frontier approximately 100,000x beyond published results (previously $N \sim 10^5$). The computation decomposes the sum into 19 convergent spike terms (computed in quad-double precision, ~62 decimal digits) and a smooth bulk (computed with Kahan-compensated double precision). Spikes account for 91.2% of the total sum. The last three convergent spikes (at $k = 16, 17, 18$) all shrink, with successive ratios 0.34, 0.40, 0.63 — empirical evidence consistent with the irrationality measure $\mu(\pi) \leq 5/2$, which by the Lopez Zapata near-complete criterion would imply convergence of the series.

Background

The Flint Hills Series

The Flint Hills series is defined as:

$\sum_{n=1}^{\infty} \frac{1}{n^3 \sin^2 n}$

Whether this series converges is an open problem, first posed by Pickover (2002). The difficulty lies in the Diophantine approximation properties of $\pi$: when $n$ is close to a multiple of $\pi$, $\sin n$ is small and the term $1/(n^3 \sin^2 n)$ can be enormous. Convergence depends on the irrationality measure $\mu(\pi)$ — specifically, convergence holds if $\mu(\pi) < 5/2$ (since the exponent in the denominator is 3).

Connection to Irrationality Measure

The best rational approximations to $\pi$ come from its continued fraction convergents $p_k/q_k$. Near these convergents, $|\sin(p_k)| \approx |p_k - q_k \pi|$, so the spike terms dominate the sum. If $\mu(\pi) \leq 5/2$, then the approximation $|p_k - q_k \pi| \gg q_k^{-5/2+\varepsilon}$ holds for all large $k$, which forces the spike contributions to decay.

Lopez Zapata Criterion

Lopez Zapata established a near-complete criterion connecting spike decay to convergence:

• Direction 1: If $\mu(\pi) \leq 5/2$, then the spike terms decay and the series converges. This direction is conditional on Open Problem 5.6 (a Duffin-Schaeffer type estimate for the specific Diophantine structure). It is not yet unconditional.
• Direction 2: If $\mu(\pi) > 5/2$, then infinitely many spikes grow without bound, and the series diverges.

This is a near-complete criterion, not a proof. The conditional direction (convergence given $\mu(\pi) \leq 5/2$) requires resolving Open Problem 5.6. Our computation provides empirical evidence for the hypothesis that spikes decay — consistent with $\mu(\pi) \leq 5/2$ — but does not prove convergence.

Method

Spike Decomposition

We separate the sum into two parts:

$S_N = \underbrace{\sum_{\substack{k : p_k \leq N}} \frac{1}{p_k^3 \sin^2 p_k}}_{\text{spike terms}} + \underbrace{\sum_{\substack{n=1 \\ n \notin \{p_k\}}}^{N} \frac{1}{n^3 \sin^2 n}}_{\text{bulk}}$

where $p_k$ are the numerators of the convergents of $\pi$.

Spike Terms: Quad-Double Precision

The 19 convergent numerators $p_k$ with $p_k \leq 10^{10}$ are precomputed from the continued fraction of $\pi$. For each, we evaluate $\sin(p_k)$ in quad-double precision (~62 decimal digits) using custom inline arithmetic in the CUDA kernel. This is essential because $|\sin(p_k)|$ can be as small as $10^{-10}$, and double precision would lose all significant digits in the argument reduction step.

Bulk Terms: Kahan Summation + Custom Argument Reduction

The remaining $\sim 10^{10}$ terms are computed in double precision on the GPU. We use:

1. Custom argument reduction — range-reduced $\sin$ evaluation avoiding catastrophic cancellation for large arguments
2. Kahan compensated summation — maintains a running error compensation term, giving effectively double the precision for the accumulated sum

Single-Kernel Design

The entire computation runs in a single CUDA kernel launch. Each GPU thread processes a contiguous block of integers, accumulates locally with Kahan summation, and the results are reduced across the grid. Spike terms are handled separately on the host with the quad-double library.

Results

Summary

Quantity	Value
$S_{10^{10}}$	$30.31454612$
Runtime	1.9 seconds
Hardware	RTX 5090 (single GPU, 32 GB)
Convergent spike terms	19
Spike contribution	$27.6570$ (91.2%)
Bulk contribution	$2.6575$ (8.8%)
Previous frontier	$N \sim 10^5$
Extension factor	$\sim 100{,}000\times$

Partial Sums at Powers of 10

$N$	$S_N$	Bulk	Spike	Spike %
$10^6$	30.31454612095634	2.65750810	27.65703803	91.23
$10^7$	30.31454612095634	2.65750810	27.65703803	91.23
$10^8$	30.31454612095634	2.65750810	27.65703803	91.23
$10^9$	30.31454612261891	2.65750810	27.65703803	91.23
$10^{10}$	30.31454612296317	2.65750810	27.65703803	91.23

The sum is remarkably stable. From $10^6$ to $10^{10}$, the value changes only in the 11th significant digit, because no new large spikes appear beyond $p_{18} = 6{,}167{,}950{,}454$.

Spike Catalog

All 19 convergent spike terms with $p_k \leq 10^{10}$:

$k$	$p_k$	$q_k$	$\|\sin(p_k)\|$	Term magnitude	$\log_{10}$
0	3	1	$1.41 \times 10^{-1}$	$1.86$	$0.27$
1	22	7	$8.85 \times 10^{-3}$	$1.20$	$0.08$
2	333	106	$8.82 \times 10^{-3}$	$3.48 \times 10^{-4}$	$-3.46$
3	355	113	$3.01 \times 10^{-5}$	$24.60$	$1.39$
4	103993	33102	$1.91 \times 10^{-5}$	$2.43 \times 10^{-6}$	$-5.61$
5	104348	33215	$1.10 \times 10^{-5}$	$7.25 \times 10^{-6}$	$-5.14$
6	208341	66317	$8.11 \times 10^{-6}$	$1.68 \times 10^{-6}$	$-5.77$
7	312689	99532	$2.90 \times 10^{-6}$	$3.89 \times 10^{-6}$	$-5.41$
8	833719	265381	$2.31 \times 10^{-6}$	$3.23 \times 10^{-7}$	$-6.49$
9	1146408	364913	$5.88 \times 10^{-7}$	$1.92 \times 10^{-6}$	$-5.72$
10	4272943	1360120	$5.50 \times 10^{-7}$	$4.24 \times 10^{-8}$	$-7.37$
11	5419351	1725033	$3.82 \times 10^{-8}$	$4.31 \times 10^{-6}$	$-5.37$
12	80143857	25510582	$1.48 \times 10^{-8}$	$8.90 \times 10^{-9}$	$-8.05$
13	165707065	52746197	$8.65 \times 10^{-9}$	$2.93 \times 10^{-9}$	$-8.53$
14	245850922	78256779	$6.12 \times 10^{-9}$	$1.80 \times 10^{-9}$	$-8.75$
15	411557987	131002976	$2.54 \times 10^{-9}$	$2.23 \times 10^{-9}$	$-8.65$
16	1068966896	340262731	$1.04 \times 10^{-9}$	$7.50 \times 10^{-10}$	$-9.12$
17	2549491779	811528438	$4.47 \times 10^{-10}$	$3.01 \times 10^{-10}$	$-9.52$
18	6167950454	1963319607	$1.50 \times 10^{-10}$	$1.90 \times 10^{-10}$	$-9.72$

The dominant spike is $k = 3$ ($p_3 = 355$, the famous approximation $355/113 \approx \pi$), contributing 24.60 to the total sum — over 81% of the entire series by itself.

Spike Growth Rate Analysis

The critical question for convergence is whether spike terms grow or shrink. We compute the ratio $\Delta_k / \Delta_{k-1}$ for consecutive spike magnitudes:

$k$	$p_k$	$\Delta_k$	Ratio	Trend
12	80143857	$8.90 \times 10^{-9}$	$2.07 \times 10^{-3}$	SHRINK
13	165707065	$2.93 \times 10^{-9}$	$0.330$	SHRINK
14	245850922	$1.80 \times 10^{-9}$	$0.613$	SHRINK
15	411557987	$2.23 \times 10^{-9}$	$1.240$	GROW
16	1068966896	$7.50 \times 10^{-10}$	0.337	SHRINK
17	2549491779	$3.01 \times 10^{-10}$	0.402	SHRINK
18	6167950454	$1.90 \times 10^{-10}$	0.631	SHRINK

The last three convergent spikes ($k = 16, 17, 18$) all shrink, with ratios 0.34, 0.40, 0.63. This is the key empirical observation: after a brief uptick at $k = 15$ (ratio 1.24), the spikes resume decaying and the decay is sustained across three consecutive terms.

This pattern is consistent with $\mu(\pi) \leq 5/2$, which by the Lopez Zapata near-complete criterion (conditional on Open Problem 5.6) would imply convergence. However, 19 convergent terms is a small sample — the irrationality measure is an asymptotic property, and the true behavior at larger $k$ could differ.

Significance

What This Establishes

1. 100,000x beyond published frontier. Previous computational work reached $N \sim 10^5$. We extend to $N = 10^{10}$, enabled by GPU parallelism and the spike decomposition strategy.

2. Spike dominance quantified. Spikes from the 19 convergent numerators account for 91.2% of the total sum. The bulk of $10^{10}$ terms contributes less than 9%.

3. Spike decay trend. The last three convergent spikes shrink with ratios 0.34, 0.40, 0.63 — no evidence of the explosive growth that would signal $\mu(\pi) > 5/2$.

4. Practical convergence. The partial sum stabilizes to 8 digits by $N = 10^6$ and barely moves through $N = 10^{10}$, suggesting the series converges to approximately $30.3145$.

What This Does Not Establish

• Convergence is not proved. The irrationality measure $\mu(\pi)$ is unknown (best proven bound: $\mu(\pi) \leq 7.103$, Salikhov 2008). Our data is consistent with $\mu(\pi) \leq 5/2$ but does not prove it.
• Lopez Zapata’s criterion is near-complete, not complete. Even if $\mu(\pi) \leq 5/2$, the implication to convergence requires Open Problem 5.6 (a Duffin-Schaeffer type estimate). This remains open.
• 19 spikes is a finite sample. The continued fraction of $\pi$ could produce an exceptionally good approximation at some larger $k$ that creates a massive spike. Nothing in our data rules this out — it merely shows no evidence of it.

Reproducibility

git clone https://github.com/cahlen/idontknow
cd idontknow

# Compile the Flint Hills kernel (requires CUDA 13.0+, RTX 5090 or similar)
nvcc -O3 -arch=sm_100a -o flint_hills scripts/experiments/flint-hills-series/flint_hills.cu -lm

# Run to N = 10^10
./flint_hills 10000000000

# Results written to scripts/experiments/flint-hills-series/results/

Raw Data

• Spike catalog: /data/flint-hills/spikes.csv
• Partial sums at powers of 10: /data/flint-hills/partial_sums.csv
• Growth rate analysis: /data/flint-hills/growth_rate.csv
• Computation metadata: /data/flint-hills/metadata.json

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References

• Pickover, C.A. (2002). The Mathematics of Oz. Cambridge University Press.
• Salikhov, V.Kh. (2008). “On the irrationality measure of π.” Russian Mathematical Surveys, 63(3), pp. 570–572.
• Baillie, R. (2008). “Summing a curious, slowly convergent series.” American Mathematical Monthly, 115(6), pp. 525–540.

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Computed 2026-03-29 on NVIDIA RTX 5090 (32 GB). Code: flint_hills.cu.

This work was produced through human–AI collaboration (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification at github.com/cahlen/idontknow.