Number theory · existing Mathlib declaration

Divergence of the Reciprocal-Prime Series

The real family that assigns the reciprocal 1/p to every prime p cannot have a finite sum.

reciprocals of primes
nonsummable families
smooth numbers
rough numbers
vanishing tails

The theorem at a glance

Divergence of Prime Reciprocals — a nonvanishing tail

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

An ivory number-theory poster states nonsummability over Nat.Primes, then shows smooth-and-rough counting feeding a prime-tail basin and a second remote cutoff with another retained basin beyond it. — Smooth and rough number counts force a finite reciprocal-prime tail of at least one half beyond every cutoff, contradicting tail vanishing under Summable.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

The corrected Scientific master is a distinct flat technical diagram. All eight required labels are exact and appear once. A finite dot field splits into k-smooth and k-rough counts; rough tokens charge across cutoff k to prime facets and terms 1/p; the resulting finite tail at least 1/2 conflicts with the tails-vanish consequence of Summable and ends at NOT SUMMABLE. The misleading graph and connecting arrow were removed, and the red contradiction bracket now spans both incompatible tail bars. No ordered partial-sum limit or quantitative rate appears. Safe margins, original-resolution legibility, RGB mode, and 1536 by 1024 dimensions pass. — Beyond every cutoff, a finite block of primes still carries a fixed positive reciprocal mass, contradicting the vanishing tails required by summability.

Complementary intuition

A mathematical landmark

Euler's theorem that the sum of prime reciprocals diverges is a classical bridge between elementary series and the distribution of primes. Mathlib retains a particularly visual Erdős proof in which smooth and rough number counts force a uniform lower bound on arbitrarily remote finite prime tails.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

theorem Nat.Primes.not_summable_one_div : ¬ Summable (fun p : Nat.Primes ↦ (1 / p : ℝ))

Selected evidence declaration text

theorem Nat.Primes.not_summable_one_div : ¬ Summable (fun p : Nat.Primes ↦ (1 / p : ℝ))

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's order-independent subtype form: the real-valued family p ↦ 1/p on Nat.Primes is not Summable. The declaration is not literally phrased as convergence of ordered partial sums to +∞, and it supplies no divergence rate or asymptotic estimate.

The exact endpoint is nonsummability over the subtype Nat.Primes, independent of any chosen enumeration.
The declaration is not literally a statement that one ordered sequence of partial sums tends to +∞.
It gives no quantitative divergence rate, asymptotic formula, or estimate for the prime-counting function.
ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics.

Provenance

Origin and evidence stay separate

Existing declaration: Nat.Primes.not_summable_one_div in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.divergence-prime-reciprocal-series.v001
Source handling: Verified upstream reference; no mirrored source package