Strip Arithmetic II · Analytic Number Theory

Slope, Width, and the Singular Series

Sequel to Strip Arithmetic I — The Goldbach Diagonal

One residue count runs the whole family. It produces min(n+1, p) in C(n), and it produces the Goldbach factor ∏(p−1)/(p−2). Same rule, two slopes — and where it saturates, the two normalisations built on it part company completely.

The number theory below is classical — reduced residues, Euler products, and the Hardy–Littlewood singular series. Nothing here is claimed as new. What is offered is a single geometric setting in which these familiar densities are one computation, and a browser that recomputes each of them from scratch.

§1 · SetupBlocks, strips, and the residue count▾

A block $\{M, M+1, \dots, M+n\}$ is $n+1$ adjacent diagonals of the coprimality lattice. Read that way, the gap diagonal identity is the width-1 case, block-coprime density $C(n)$ is the general width case, and turning the slope from $+1$ to $-1$ gives the configuration Hardy and Littlewood treat for Goldbach. Each of the three is a standard piece of elementary or classical analytic number theory. What the lattice adds is that they are visibly the same local computation, run at different slopes and widths.

Most of what follows is one walk over a strip: the set of lattice points $(r, M)$ satisfying

$$ s \;\le\; b\,r - a\,M \;<\; s + w $$

for a slope $a/b$, an offset $s$, and a width $w$. The block case is the exception, and it is worth stating precisely because getting it wrong is easy: $C(n)$ is not the coprime density inside a diagonal strip. The block $\{M,\dots,M+n\}$ runs along $M$, and the condition is a conjunction — the point $r$ must be coprime to every member:

$$ \gcd(r,\,M+j) = 1 \quad \text{for all } j = 0,\dots,n $$

A vertical band of $n+1$ columns with an AND across it, not an average within a diagonal strip. The density of cells surviving that AND is $D(n)$; the average coprime density inside a width-$(n{+}1)$ diagonal strip is a different and much larger number. Band mode below measures the first. This is the same condition Strip Arithmetic I applies in its block n control, walked over the whole lattice here instead of along a single strip.

Slope	Width	What the strip is	Exact density
`+1`	`1`	Gap diagonal $r - M = g$	$\varphi(g)/g$
`vertical`	`n+1`	Block of $n{+}1$ consecutive integers, AND across the band	$D(n) = \prod_p\!\left(1 - \tfrac{\min(n+1,p)}{p^2}\right)$
`−1`	`1`	Goldbach anti-diagonal $r + M = N$ — Strip I	$\varphi(N)$ points exactly
`a/b`	`1`	Farey sector ray	Stern–Brocot mediant structure
`+1` or `−1`	`1`	The same strip read along its index $k$ rather than along the lattice	first zero at $\mathrm{spf}(k)$ — so $\pi(x)$ exactly, over a finite range

Rows one, three and four are the same statement at $w=1$. On any line $b\,r - a\,M = c$ a prime can only kill a cell if it divides $c$, so that line's density is $\prod_{p \mid c}(1-1/p) = \varphi(|c|)/|c|$, and a width-$w$ band is the weighted mean of those over its $w$ lines. That is a closed form for every slope and width. But it is a mean of per-line densities, where $D(n)$ in row two is a product of local factors — two exact answers to two different questions, which is what the Walk selector picks between.

The rule underneath

Fix a finite set of integer offsets $\mathcal{H}$. For each prime $p$, let

$$ \nu_p(\mathcal{H}) \;=\; \#\{\, h \bmod p \;:\; h \in \mathcal{H} \,\} $$

be the number of distinct residue classes $\mathcal{H}$ occupies mod $p$. Every local factor on this page is a function of $\nu_p$ and nothing else. Where two offsets collide mod $p$, $\nu_p$ drops by one and the local factor relaxes by a corresponding amount.

Slope +1 — the block

$\mathcal{H} = \{0,1,\dots,n\}$. Consecutive integers, so by pigeonhole $\nu_p = \min(n+1, p)$ — every class is hit once $p \le n+1$. That is exactly the saturation mechanism: the local factor freezes at $1 - 1/p$ and stops moving.

Slope −1 — Goldbach

$\mathcal{H} = \{0, N\}$, since $r$ and $N-r$ must both be prime. So $\nu_p = 2$ — unless $p \mid N$, when the two offsets fall in the same class and $\nu_p = 1$. The same collision as the block case, arrived at by division rather than by pigeonhole.

Two normalisations, and where they part company

One count, but two ways of turning it into a density — and the difference between them is the single most useful thing on this page.

$$ D(n) \;=\; \prod_p \left(1 - \frac{\nu_p}{p^2}\right) \qquad\qquad \mathfrak{S}(\mathcal{H}) \;=\; \prod_p \frac{1 - \nu_p(\mathcal{H})/p}{(1 - 1/p)^{k}} $$

$D$ asks how often one integer is coprime to the whole of $\mathcal{H}$, against no baseline at all. $\mathfrak{S}$ asks how often every member of $\mathcal{H}$ is prime at once, measured against what the prime density alone would predict — hence the $(1-1/p)^{k}$ in the denominator, $k = |\mathcal{H}|$. Same $\nu_p$ on the way in, two different questions on the way out.

Admissibility — one event, two meanings

Classically $\mathcal{H}$ is admissible when $\nu_p(\mathcal{H}) < p$ for every prime $p$. Otherwise $\mathcal{H}$ covers a full residue class system mod $p$, the factor $1 - \nu_p/p$ vanishes, and $\mathfrak{S}(\mathcal{H}) = 0$. The consecutive block $\mathcal{H} = \{0,\dots,n\}$ fails this for every $p \le n+1$ — precisely the range where $\nu_p = \min(n+1,p) = p$.

So saturation and inadmissibility are the same event. It is fatal for $\mathfrak{S}$: that product asks whether the whole tuple can be prime at once, and a covered prime forbids it outright. It is harmless for $D(n)$, whose factor merely settles at $1 - 1/p$, because coprimality of one number to a block stays perfectly possible. Two normalisations, two meanings of $\nu_p = p$ — which is why a block and a prime pair, indistinguishable as strips in the lattice, behave nothing alike.

The rest of this section follows the two branches in turn, and §4 keeps their statuses apart.

The $\mathfrak{S}$ branch — slope −1

The singular series was introduced in Partitio Numerorum III (1923). Everything on this page sits at $k=2$, since the Goldbach condition is the pair $\mathcal{H}=\{0,N\}$, which is admissible for every $p$. Evaluating the product there is finite algebra and the classical closed form follows:

$$ \mathfrak{S}(N) \;=\; 2\,C_2 \prod_{\substack{p \mid N \\ p > 2}} \frac{p-1}{p-2}, \qquad C_2 = \prod_{p>2}\left(1 - \frac{1}{(p-1)^2}\right) \approx 0.6601618158 $$

The $p = 2$ collision supplies the leading $2$. Every odd prime dividing $N$ supplies one $\frac{p-1}{p-2}$. The correction is not an extra ingredient: it is the collision count, read off the same $\nu_p$ that gives $\min(n+1,p)$ in the block case. The algebra is nine lines and is worth following once.

▾Derivation — how the product collapses, step by step

Fix the residue count. With $\mathcal{H}=\{0,N\}$: the two offsets are distinct mod $p$ unless $p \mid N$. So $\nu_p = 2$ for $p \nmid N$, and $\nu_p = 1$ for $p \mid N$.
Split the product on that dichotomy. $$\mathfrak{S}(N) = \prod_{p \mid N} \frac{1 - 1/p}{(1-1/p)^2} \;\cdot\; \prod_{p \nmid N} \frac{1 - 2/p}{(1-1/p)^2}$$
The divisor factors cancel down. One power of $(1-1/p)$ divides out: $$\frac{1-1/p}{(1-1/p)^2} = \frac{1}{1-1/p} = \frac{p}{p-1}$$
Handle $p=2$ first, because it is the only factor that could vanish. For the non-divisor factor at $p=2$, $1 - 2/p = 0$. That factor never appears: $N$ is even, so $2 \mid N$ always, and $p=2$ sits in the divisor branch, contributing $\frac{2}{2-1} = 2$. This is where the leading $2$ comes from — not from a convention, but from the fact that a Goldbach $N$ is even.
Rewrite the non-divisor factor for odd $p$. $$\frac{1-2/p}{(1-1/p)^2} = \frac{p(p-2)}{(p-1)^2} = \frac{p^2-2p+1-1}{(p-1)^2} = 1 - \frac{1}{(p-1)^2}$$ which is exactly the factor in Hardy and Littlewood's $C_2$.
Complete the product over all odd $p$. The product over $p \nmid N$ is the product over all odd $p$ divided by the terms for $p \mid N$: $$\prod_{\substack{p \nmid N \\ p>2}}\!\left(1 - \tfrac{1}{(p-1)^2}\right) = C_2 \Big/ \prod_{\substack{p \mid N \\ p>2}}\!\left(1 - \tfrac{1}{(p-1)^2}\right)$$ This is the step that produces $C_2$: it is what is left when the divisors are removed from a product that is otherwise over every odd prime.
Recombine the two branches, prime by prime. Each odd $p \mid N$ now carries the divisor factor from step 3 and the reciprocal from step 6: $$\frac{p}{p-1} \cdot \frac{(p-1)^2}{p(p-2)} = \frac{p-1}{p-2}$$
Collect. The $p=2$ branch gives $2$, the completed odd product gives $C_2$, and each odd $p \mid N$ gives $\frac{p-1}{p-2}$: $$\mathfrak{S}(N) = 2\,C_2 \prod_{\substack{p \mid N \\ p>2}} \frac{p-1}{p-2}$$
Read the result. $\frac{p-1}{p-2} > 1$ for every odd $p$, so $\mathfrak{S}$ is raised by every odd prime factor, and raised most by the small ones ($p=3$ gives $2$; $p=97$ gives $1.011$). An $N$ divisible by $3$, $5$ and $7$ — say $N=210$ — carries $2 \cdot \tfrac{4}{3} \cdot \tfrac{6}{5} = 3.2$, while a prime-poor neighbour like $N = 206 = 2 \cdot 103$ carries only $\tfrac{102}{101} \approx 1.01$. Both are multiplied by the same $2C_2$, so $\mathfrak{S}(210) \approx 4.23$ against $\mathfrak{S}(206) \approx 1.33$. That ratio, not the size of $N$, is what splits the comet into bands.

This is the standard evaluation, reproduced here rather than asserted; every equality is finite algebra on a convergent product. What is conjectural is not the closed form but the asymptotic that uses it — that $\mathfrak{S}(N)\,\mathrm{li}_2(N)$ approximates $r(N)$ at all. §4 separates the two.

The $D$ branch — the block

The other normalisation, on the tuple that is maximally inadmissible. Nothing here is conjectural: the local count is exact, the combination across primes is CRT, and the density statement is Möbius inversion.

▾Derivation — where $D(n) = \prod_p (1 - \nu_p/p^2)$ comes from

State the event. A cell $(r,M)$ survives when $\gcd(r, M+j) = 1$ for every $j = 0,\dots,n$. It fails when some prime $p$ divides $r$ and divides at least one member of the block.
Work one prime at a time. Fix $p$ and treat $r$ and $M$ as uniform and independent mod $p$. Then $p \mid r$ with probability $1/p$.
Count how much of the block $p$ can hit. Given $p \mid r$, the failure needs $M + j \equiv 0$ for some $j \le n$, i.e. $-M$ must land in the set $\mathcal{H} = \{0,\dots,n\}$ reduced mod $p$. That set occupies $\nu_p = \min(n+1, p)$ classes, so this happens with probability $\nu_p/p$.
Multiply for the local failure rate. The two conditions are independent, so $p$ spoils the cell with probability $\tfrac{1}{p}\cdot\tfrac{\nu_p}{p} = \nu_p/p^2$, and the local survival factor is $1 - \nu_p/p^2$.
Multiply across primes. Conditions at distinct primes are independent by CRT, giving $$D(n) = \prod_p \left(1 - \frac{\nu_p}{p^2}\right)$$
Read the saturation. Once $p \le n+1$ the block covers every class, $\nu_p = p$, and the factor becomes $1 - p/p^2 = 1 - 1/p$. It cannot fall further no matter how much $n$ grows. That is why $C(n)$ flattens: the small primes are already fully committed, and only primes above $n+1$ are still contributing anything new.

The probabilistic wording in steps 2–4 is shorthand for an exact count: mod $p$ there are $p^2$ residue pairs $(r,M)$ and exactly $\nu_p$ of them fail, and the combination across finitely many primes in step 5 is the Chinese Remainder Theorem, not an assumption. Made formal, the statement is the usual Möbius inversion over squarefree $d$, $$\#\{(r,M)\in[1,X]^2 : \gcd(r,M+j)=1\ \forall j\} \;=\; D(n)\,X^2 + o(X^2),$$ with the tail of the Euler product supplying the error — the same argument that gives $6/\pi^2$ for the density of coprime pairs when $\mathcal{H}=\{0\}$. So $D(n)$ is a theorem of the elementary kind, not a heuristic; the self-test panel below measures it against a direct walk anyway.

§2 · The latticeStrips at any slope and width

Live identity check — measured against exact▾

Quantity	Measured on the strip	Exact	Δ

Slope $+1$, width 1, offset $g$: the strip density must equal $\varphi(g)/g$. Slope $-1$, width 1, offset $N$: the point count must equal $\varphi(N)$ exactly — this is the count behind Strip Arithmetic I, where the anti-diagonal is fully coprime iff $\varphi(N) = N-1$ — a restatement of the elementary criterion that $\varphi(N) = N-1$ characterises the primes.

Prime counter — the first zero on a strip▾

A strip carries an index. For the gap diagonal it is $g$ and the cell at depth $M$ is $\gcd(g, M)$; for the Goldbach anti-diagonal it is $N$ and the cell at $r$ is $\gcd(N, r)$. Those are the same scan — coprimality to a fixed integer $k$ — so both slopes give one picture, and two exact facts fall out of it:

The first zero along the strip sits at $\mathrm{spf}(k)$, the smallest prime factor. Not approximately: exactly, every time.
Below it the zeros repeat with period $\mathrm{rad}(k)$, which is the periodic block structure the plate shows. Their count to depth $K$ is $K - \sum_{d \mid \mathrm{rad}(k)} \mu(d)\lfloor K/d \rfloor$.

So a strip clean to depth $K$ means $k$ has no prime factor $\le K$, and for $k \le K^2$ that forces $k$ prime. Counting clean strips to depth $K = \lfloor\sqrt{x}\rfloor$ gives $\pi(x) - \pi(K)$ — the sieve of Eratosthenes in strip coordinates, with Legendre's inclusion–exclusion as the same Möbius argument §1 already uses. Three routes are computed below by three different implementations and must agree.

Range x

depth K = 70

Window k = 2 … click a column to pick a strip — the rest dim

The picture is a window at one column per strip, so it is exact rather than aggregated. The counts below cover the whole range.

gcd distribution — counts, ratios, and the exact form▾

Over the square $[1,n]^2$, the number of ordered pairs with $\gcd(r,M)=d$ is exactly $\Phi(\lfloor n/d \rfloor)$, where $\Phi(m) = 2\sum_{k\le m}\varphi(k) - 1$ counts the coprime pairs in $[1,m]^2$. Two identities fall out and both are checked live below: $\sum_d \Phi(\lfloor n/d\rfloor) = n^2$ exactly, and $\Phi(n)/n^2 \to 6/\pi^2$.

Sample n

rows

Self test — every stated identity, recomputed in the browser▾

Nothing above is taken on trust. Each row recomputes a stated identity from scratch and compares it against an independent route — a bitset against direct gcd, a sieve against a closed form, a convolution against a published constant. Red is a real failure.

not run

§3 · The cometThe singular series as the comet’s band structure▾

The Goldbach comet plots $r(N)$, the number of ordered prime pairs summing to $N$, against $N$. It does not form a single curve: it separates into distinct bands. That the bands are indexed by $\mathfrak{S}(N)$, and that dividing by $\mathfrak{S}(N)$ merges them, is the standard reading of the comet and dates to the same 1923 paper. The panels below reproduce that normalisation on data computed in the browser. They illustrate a classical prediction; they are not an argument for it, and §4 keeps the two apart.

▾What $\mathrm{li}_2(N)$ is, and one caveat about the tables below

The tables report $r(N)$ divided by a prediction, so the prediction needs stating. Under the heuristic that an integer near $t$ is prime with density $1/\log t$, the expected number of ordered pairs summing to $N$ is $\mathfrak{S}(N)\,\mathrm{li}_2(N)$, where

$$ \mathrm{li}_2(N) \;=\; \int_2^{N} \frac{dt}{(\log t)^2} $$

computed here by composite Simpson, split at $t=100$ because the integrand falls from about $2.08$ to $0.047$ across that interval and one panel width cannot serve both ends. The routine is checked against a 400,000-step reference at two points.

The caveat. The pair really being counted is $(a, N-a)$, whose natural density is $\frac{1}{\log a \log (N-a)}$, giving $\int_2^N \frac{dt}{\log t \,\log(N-t)}$. The single-log form used here is the standard stand-in and agrees with it asymptotically, but the two differ at relative order $1/\log N$. That is the most likely reason the with $\mathfrak{S}$ column sits near $0.95$–$0.97$ rather than $1.00$ — a systematic, size-dependent deficit, not a failure of $\mathfrak{S}$. The column to read for whether $\mathfrak{S}$ works is the spread across neighbouring $N$, not the distance of any single row from $1$.

Comet parameters

▾Controls — N max, view mode

N max

60000

View point α .35

——

Export

Spread of r(N) / (𝔖·li₂) — the number that matters▾

If $\mathfrak{S}$ is doing real work, the spread of the normalised ratio should be far tighter than the spread of the unnormalised one. Computed live over every even $N$ in range.

Normalisation	mean	std dev	min	max	max/min

Neighbouring N, very different radicals▾

A window of even $N$ centred on the one you choose. All of them are within a few dozen of each other, so any size-dependent effect is constant across the window — what is left is the radical, which varies by orders of magnitude from row to row. The without column moves with it; the with $\mathfrak{S}$ column does not. The live spread figure beside the controls is the ratio of largest to smallest without value across the window, which is the number the correction has to remove.

centre N

rows either side

—

N	rad(N)	r(N)	𝔖(N)	with 𝔖	without

§4 · StatusProvenance of each statement▾

Nothing on this page is offered as a new theorem. The material is classical: elementary counts, one classical theorem, one classical evaluation, and one conjecture that is still open. Open this section for which register each statement sits in — and for what the page does not claim.

The material divides into three classical registers, and the point of the section is to say which register each statement belongs to.

Elementary, classical. $\varphi(g)/g$ on the $+1$ diagonal and $\varphi(N)$ on the $-1$ anti-diagonal. Both are counts of reduced residues; they are checked here by direct enumeration rather than derived again.
Elementary, classical. $\nu_p(\mathcal{H}) = \min(n+1,p)$ for a consecutive block (pigeonhole) and $\nu_p \in \{1,2\}$ for $\mathcal{H} = \{0,N\}$ (whether $p \mid N$). Finite residue counts, verified above.
Classical theorem. $D(n) = \prod_p (1 - \nu_p/p^2)$ as the natural density of the surviving pairs $(r,M)$, by Möbius inversion and CRT — the $\mathcal{H}=\{0\}$ case is the familiar $6/\pi^2$. The $C(n) = \zeta(2)\,D(n)$ normalisation is bookkeeping, not content.
Classical evaluation. The collapse of $\prod_p \frac{1-\nu_p/p}{(1-1/p)^2}$ to $2C_2\prod_{p\mid N,\,p>2}\frac{p-1}{p-2}$ is Hardy and Littlewood's, Partitio Numerorum III (1923). Reproduced here for legibility.
Elementary, classical — and exact over a finite range. The first zero on the strip indexed by $k$ is at $\mathrm{spf}(k)$, so counting strips clean to depth $\lfloor\sqrt{x}\rfloor$ gives $\pi(x) - \pi(\lfloor\sqrt{x}\rfloor)$ with no error term and no conjecture. This is the sieve of Eratosthenes in strip coordinates; the inclusion–exclusion form is Legendre's. No claim of novelty or of speed: Meissel, Lehmer, and Lagarias–Miller–Odlyzko compute $\pi(x)$ far faster, and Legendre's $2^{\pi(y)}$ term is exactly why. It counts a finite range and says nothing as $x \to \infty$.
Conjectural. That $\mathfrak{S}(N)\,\mathrm{li}_2(N)$ approximates $r(N)$. This is the Hardy–Littlewood asymptotic, open since 1923. The comet panels display it; display is not support.
Outside the scope entirely. Goldbach itself. The lattice resolves coprimality; primality is a strictly harder condition and nothing here approaches it.
What the page supplies. A presentation, not a result: one geometric setting (slope, width, offset) and one parameter ($\nu_p$) under which the four rows of the table in §1 are the same computation, together with browser-side arithmetic that recomputes each of them independently.

Normalisation note

$C(n)$ uses local factors $1 - \nu_p/p^2$ because it averages over block anchors — it is a density of integers. The singular series uses $\frac{1-\nu_p/p}{(1-1/p)^2}$ because it compares against a prime-density baseline. Different normalisations of the same $\nu_p$. Conflating them would be an error; the shared object is the residue count, not the Euler factor.

Strip Arithmetic II · Wessen Getachew · 2026
Part II of the strip series — Strip Arithmetic I: The Goldbach Diagonal.
Companion to Page 5 — Gap Diagonal Identity, Page 6 — Strip Correlations and Page 8 — k-Block Density.
The strip framing and the interactive arithmetic are this page's; the number theory is not. Full attribution is in the credits panel below.

Credits — whose mathematics this is▾

Almost nothing on this page originates here. What follows records who each result belongs to and how it enters, including the pieces that are load-bearing without being visible. Names appear because their work is used, not because they have any connection to this page: no one listed has reviewed it, and nothing here should be read as carrying their endorsement. Errors of attribution are mine and I would rather be told than left alone with them.

off — the list reads as prose

Open lattice credits in

Directly used — results this page states, restates, or recomputes

Whose	What it is doing here
Christian Goldbach & Leonhard Euler 1742 correspondence	The conjecture the slope $-1$ anti-diagonal encodes. Euler's reply is the reason the even-$N$ form is the one everyone states.
Leonhard Euler	$\varphi(n)$; the Euler product over primes; and $\zeta(2) = \pi^2/6$, which is the entire content of the $C(n) = \zeta(2)\,D(n)$ normalisation. $\varphi$ also supplies the strip walk's own closed form: on any line $b\,r - a\,M = c$ the density is $\varphi(\|c\|)/\|c\|$, and a band is the weighted mean of those over its lines.
August Ferdinand Möbius	$\mu(d)$ and Möbius inversion — the step that turns the local counts $1-\nu_p/p^2$ into a density statement about $D(n)$ rather than a heuristic. Separately, the Cayley view's disk map is a Möbius transformation: same name, unrelated object, and worth not confusing.
Srinivasa Ramanujan	The sums $c_M(n) = \sum_{\gcd(r,M)=1} e^{2\pi i r n/M}$. The arrow the collapsed ring draws in its void is $c_M(1) = \mu(M)$ — the same $\mu$ Möbius supplies to the $D(n)$ derivation, arriving here as a resultant rather than as a weight.
Sunzi Suanjing and the classical Chinese remainder tradition	CRT is what licenses multiplying local factors across distinct primes. It is the quietest assumption in the derivation and the one doing the most work.
Franz Mertens (1874), Ernesto Cesàro (1881–83), P. G. L. Dirichlet (1849)	The density of coprime pairs is $1/\zeta(2)$ — the $\mathcal{H}=\{0\}$ case of everything in §1, and the target the $\Phi(n)/n^2$ panel converges to. Mertens gave the Möbius proof; Cesàro reached it independently; Dirichlet's mean-value work is usually named as the origin. Chebyshev popularised the question in its “is a random fraction already in lowest terms?” form.
G. H. Hardy & J. E. Littlewood Partitio Numerorum III, Acta Math. 44 (1923)	The residue count $\nu_p(\mathcal{H})$, the singular series $\mathfrak{S}(\mathcal{H})$, admissibility, the constant $C_2$, and the conjectural asymptotic for $r(N)$. §1's collapse is their calculation and §3's normalisation is their prediction. This page is largely a rendering of that one paper.
Henry F. Fliegel & Douglas S. Robertson “Goldbach's Comet”, J. Recreational Math. 21(1), 1989	The comet plot itself, and the first published account of why it bands — a modular argument, i.e. the same observation §3 makes. The comet panel is a re-rendering of their figure with live data.
Carl Friedrich Gauss	The logarithmic integral and the $1/\log t$ density heuristic, which is what $\mathrm{li}_2(N)$ is built from.
Thomas Simpson	Composite Simpson's rule — how $\mathrm{li}_2(N)$ is actually evaluated in the browser, split at $t=100$.
Adrien-Marie Legendre; Meissel, Lehmer, Lagarias–Miller–Odlyzko	Legendre's $\Phi(x,y)$ and the inclusion–exclusion that turns clean strips into $\pi(x)$; the counter panel runs it as an independent cross-check. The others are the modern combinatorial $\pi(x)$ algorithms, named in §4 as the reason this page makes no speed claim.
Eratosthenes	Sieving by clearing multiples rather than testing. The lattice bitset is built exactly this way, and that choice is the reason $G$ can reach $10^4$ on a phone.
John Farey, Charles Haros, Moritz Stern, Achille Brocot	Mediants and the Stern–Brocot structure, cited in the slope $a/b$ row of the §1 table — the general-slope case the other three rows are special cases of.

Indirectly — the shape of the argument, borrowed

Whose	Relationship
Viggo Brun and the sieve tradition after him; Halberstam & Richert, Sieve Methods	Admissible tuples and local densities in the form used here. The habit of separating an exact local count from a conjectural global asymptotic — which is the organising idea of §4 — is sieve-theoretic and not original.
Montgomery & Vaughan, Multiplicative Number Theory I; Hardy & Wright, An Introduction to the Theory of Numbers	The modern statements this page's notation follows, and where the coprime-density theorems are found in their standard form.
J. Bureaux & N. Enriquez “The probability that two random integers are coprime” (2016)	The error term behind the $\Phi(n)/n^2 \to 6/\pi^2$ panel. This page only displays the convergence; the size of the remainder is their subject, not one this page addresses.
Arthur Cayley; Henri Poincaré	The half-plane-to-disk transform and the disk model behind the Cayley/ring projection. Presentational here rather than arithmetical — the mathematics does not depend on the view.
Tomás Oliveira e Silva	Large-scale numerical verification of Goldbach. Not used here and no computation on this page approaches that range; listed because anyone reading a comet plot should know where the real verification effort lives. no control — nothing on this page computes at that scale
N. J. A. Sloane and OEIS contributors	$C_2$ as A005597, and the cross-checking of every constant quoted on this page against a published value.

Tools

MathJax for typesetting. Typefaces via Google Fonts: Cinzel (Natanael Gama), Spectral (Production Type), JetBrains Mono (JetBrains). Everything else — lattice, sieve, projections, comet, self-tests — is hand-written vanilla JavaScript and Canvas with no framework or plotting library.

Same author

Strip Arithmetic I — The Goldbach Diagonal, the Explorer, and the C(n) suite: Page 5, Page 6, Page 8. The strip framing, the slope/width parametrisation, and the browser arithmetic are the parts of this page that are mine; they are a way of looking, not a result. These companion pages don't link back here — use your browser's back button to return.