WESSEN GETACHEW P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 ½+it P15 Opener
Analytic Number Theory · Wessen Getachew

The Prime Filter — A Sieve-of-Eratosthenes View

$$\text{density}(M,n) \;=\; \prod_{p \,\in\, \text{Touch}(M,n)} \left(1-\frac{1}{p}\right), \qquad \text{Touch}(M,n) = \{p \le n{+}1\} \,\cup\, \{\, p \mid M(M{+}1)\cdots(M{+}n),\; p > n{+}1 \,\}$$
Pages 1, 7, and 8 all average over every possible block anchor M, which is why they need an infinite Euler product. Fix a specific M instead, and the sieve becomes finite: only the primes that actually divide one of M,…,M+n can ever remove anything. Every prime p ≤ n+1 always touches (pigeonhole — among any p consecutive integers, one is divisible by p); every larger touching prime is there by coincidence, because it happens to divide one specific term. Once you've sieved out every touching prime, the sieve is done — provably, not asymptotically — and the surviving density is an exact, finite product. Verified below against direct enumeration.
Live Sieve Explorer
n (block depth) 6
M (block anchor) 100
Steps through the touching primes in increasing order, removing multiples of each from the r-grid — exactly Eratosthenes' original method, applied to this specific block.
alive (survived every prime processed so far) just removed this step removed earlier
Primes processed
0 / 0
Alive fraction (grid, N=—)
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Exact target ∏(1−1/p) over all touching primes
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M-averaged D(n) from Page 1 (for contrast)
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Touching Primes, In Sieve Order
Only these primes ever remove anything from the grid — every other prime is a complete no-op for this specific (M,n).
steppwhy it touchesfactor (1−1/p)cumulative product
Why This Sieve Is Exact (and Most Aren't)
1 — Legendre's exact sieve, and why it's usually impractical
The classical exact sieve computes |{n ∈ A : gcd(n, P(z)) = 1}| via inclusion–exclusion over every squarefree divisor of P(z) = ∏_{p<z} p — that's 2^π(z) terms. For z around a few hundred, π(z) is already large enough that direct evaluation is hopeless, which is exactly why Brun's and Selberg's sieves exist: they truncate the inclusion–exclusion and trade exactness for a provable upper or lower bound instead of an exact value.
2 — Why our sieve skips that problem entirely
We never do inclusion–exclusion at all. Fixing M makes the touching-prime set finite and explicitly computable — just factor the n+1 numbers M,…,M+n — and because divisibility by distinct primes is independent (CRT), the surviving density is a straight product over that finite set, not an alternating sum over its power set. No truncation, no bound, no error term: exact by construction.
3 — Why this doesn't generalize to hard sieve problems
Classical sieve applications (twin primes, Goldbach, prime gaps) don't get this luxury: the relevant "bad" events aren't cleanly independent across primes at every scale the way block-coprimality's are, so no finite touching-set trick is available and bounding sieves remain necessary. This project's object is unusually well-behaved, which is precisely why C(n), C(n₁,…,n_k), and this exact finite sieve are all provable outright rather than conjectural.
4 — This density is not D(n), and that's the point
Pages 1, 7, and 8 compute D(n) = C(n)/ζ(2), the density averaged over every M. This page computes the exact density for one specific M — a single "phase" of that average, not the average itself. They generally differ (see the result box above) — which is precisely why the finite-lattice canvases on Pages 7 and 8 display the fixed-M count and the M-averaged prediction side by side rather than conflating them.