Tao 2024 monotone nondecreasing sequences euler totient function

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composite_barrier: A composite placed after a prime while preserving its totient must lie beyond an explicit square-root gap.

corollary_1_2: Every weakly increasing totient subset has reciprocal sum at most log log x plus an absolute constant.

dedekind_fibre: A finite prime-support induction handles the two-three cancellation and proves the sharp Dedekind fiber bound.

external_context: Separate the source’s historical bounds, conditional comparisons and unperformed finite computations from the compiled proof chain.

half_bound: Bound the restricted monotone maximum with leading upper coefficient one-half after excluding the two extremal families.

lemma_1_5: Bound a weighted sum over smooth integers by its largest Rankin weight, with a logarithmic factor.

lemma_1_6: Derive a uniform interval prime count from the classical prime number theorem with its exponential error.

lemma_1_7: Control the reciprocal sum over a short or long prime interval uniformly in both endpoints.

lemma_2_1: Identify the unique prime support of a nonempty totient-ratio fiber and compute its reciprocal mass.

lemma_3_1: Cover the integers by a primary family, a negligible monotone family and six controlled exceptional classes, which may overlap.

notation: Fix the finite maxima, smoothness conventions and precise classical prime estimates used throughout the proof.

proposition_1_4: Every positive rational totient ratio has reciprocal fiber mass at most one.

proposition_3_2: Bound all six exceptional factorization classes by the error allowed in the main theorem.

proposition_3_3: Use local monotonicity and interval counting to make the secondary factorization family negligible.

proposition_3_4: Bound the primary family with the printed logarithmic error using exact fiber mass and a uniform logarithmic moment.

proposition_4_1: Infinitely many prime counterexamples to the stated Legendre interval force an unbounded additive excess over the primes.

proposition_4_5: A uniform shortage of prime-ceiling pairs forces an additive totient excess of order at least x over log squared x.

remark_2_2: The reciprocal fiber bound is sharp only at ratios one and one-half, and each totient ratio determines its prime support.

remark_4_6: For each positive threshold only finitely many sum-of-divisors ratio fibers have reciprocal mass above it.

remark_4_7: The weakly increasing Dedekind totient maximum has the same prime asymptotic and reciprocal bound.

strict_transfer: Transfer the weak maximum theorem to the strict asymptotic and density questions in Problem 49.

sum_of_divisors_analogue: The same leading asymptotic and reciprocal-sum bound hold for weakly increasing sums of divisors.

sum_of_divisors_fibre: Zhang’s powerful-number argument bounds each sum-of-divisors ratio fiber, with equality only at ratio one.

theorem_1_1: The largest weakly increasing totient subsequence has size asymptotic to the number of primes.

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Terence Tao, Monotone Nondecreasing Sequences of the Euler Totient Function, La Matematica 3(2) (2024), 793–820, DOI 10.1007/s44007-024-00115-z.

The canonical PDF is the published 28-page version: physical page $j$ is printed page $792+j$. The arXiv v4 edition, not held, has 23 pages and is dated 6 April 2024. The published paper records receipt on 10 September 2023, revision on 25 April 2024, acceptance on 30 April 2024 and online publication on 23 May 2024. Exact identities and primary metadata are in the source record. The arXiv record, still lists v4 as its latest version. The canonical tao_2024_monotone_nondecreasing_sequences_euler_totient_function.pdf prints "© The Author(s) 2024" on its first page and, on printed p. 819 (physical p. 27), "Open Access This article is licensed under a Creative Commons Attribution 4.0 International License" with the license address http://creativecommons.org/licenses/by/4.0/, the Creative Commons Attribution 4.0 license. For the arXiv v4 edition, the arXiv record names arXiv's non-exclusive distribution license (arXiv:2309.02325), every other right reserved.

For the weak maximum defined in notation, Theorem 1.1 proves

$$\pi(x)\le M(x)\le \left(1+O\!\left(\frac{(\log\log x)^5}{\log x}\right)\right)\pi(x) \qquad(x\ge10).$$

Corollary 1.2 gives a reciprocal-sum bound $\log\log x+O(1)$ for each such subset. The strict transfer proves the asymptotic and $o(N)$ clauses of Problem 49, whose statement uses strict totient inequalities. It does not assert that primes are an exactly largest strict example for every $N$.

The complete main chain. The proof decomposes typical integers as either $dp$ or $dps$. In the second family, local monotonicity constrains the intermediate prime and forces a negligible contribution. In the first, rational separation and the exact reciprocal mass of a totient-ratio fiber supply the leading constant one.

• Lemma 1.5 proves the smooth-number Rankin bound.
• Lemma 1.6 and Lemma 1.7 give the precise prime counts and reciprocal windows.
• Lemma 2.1 proves Proposition 1.4; Remarks 2.2–2.3 identify equality and prime support.
• Lemma 3.1 gives the full factorization decomposition.
• Proposition 3.2 bounds all six exceptional classes.
• Proposition 3.3 bounds the secondary family.
• Proposition 3.4 treats the primary family, including the logarithmic-moment expansion needed for its stated error.
• Theorem 1.1, Corollary 1.2 and the strict transfer complete the consequences.

These are ordinary proofs relative to the precise classical PNT and Mertens inputs stated in notation. The analytic proofs of those external theorems are not included.

Distinct additional arguments. Proposition 4.1 and its Oppermann variant insert bad prime squares. Their conclusion is conditional on infinitely many prime-gap counterexamples. Proposition 4.5 constructs an excess from powers of two times primes, under its explicit uniform shortage of prime-ceiling pairs. The restricted-family half bound and composite barrier are complete local deductions.

The sum-of-divisors fibre proof uses Shengtong Zhang's different powerful-number argument. It also yields finiteness of the large fibres, and the full sigma analogue supplies the same main asymptotic and reciprocal bound. In arXiv v1 the sharp sigma inequality was still a missing input. Tao's dated blog discussion records Zhang's contribution in October 2023; v3 and the later versions include it. The Dedekind fibre induction and Remark 4.7 fully supply the finite cases and adaptations left to the reader in the published source. Tao attributes this Dedekind extension to an anonymous referee.

Source precision. The result pages preserve the relevant issues: the reversed subtraction in the introductory motivation; the strict endpoint in Lemma 3.1; the extra fiber logarithmic moment for the stronger Proposition 3.4 error; the larger post-deletion remainder in Proposition 4.5; and the $q>1$ range and finer mesh required by sigma and psi. These are explicit compilation expansions, not an author-issued erratum. The source's dyadic Vinogradov bound and its hull-order direction are correct and are not treated as errors.

Scope limits. external_context separates historical constant and decreasing maxima, the finer additive-64 conjecture, the unspecified finite computation in Remark 4.3, and the exact external Selberg/RH and Maynard comparisons. The latter are not main-proof inputs or new complete proof claims here. Source-dated questions are not assertions of current openness. This unit does not certify any formal proof, public build or local kernel verification.

Bears on. Problem 49.

Only the edition under an open license is held; the source's other editions are not, since no license on record permits their redistribution, and the card cites the edition it names above.