Lagarias 2006 annotated bibliography 2

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conjecture_p1: The 3x+1 Conjecture as the second bibliography's abstract and introduction pose it, for the 3x+1 function T, which is the map of Problem 1135, with the Collatz function C beside it, the verification bounds the introduction reports, and the conjecture recorded as unsolved.

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Jeffrey C. Lagarias, The 3x+1 Problem: An Annotated Bibliography, II (2000-2009), arXiv:math/0608208v6 (2012 revision; 42 pp.).

Relevance: Lagarias's annotated bibliography of the 3x+1 literature 2000-2009, the second installment after the bibliography of 1963-1999, covering work on problem 1135.

One annotated entry per work, sorted by author; the bibliography proves nothing of its own. Its abstract and the opening of §1 (p. 1; §1 runs to p. 2) define the $3x+1$ function $T$ and the Collatz function $C$, pose the $3x+1$ Conjecture and report the verification bounds (one as of Feb. 21, 2008), recorded on the conjecture page; §2 (p. 2) fixes the terminology (trajectory, stopping times, height and the counting function $\pi_a(x)$) used in the entries.

The copy read for this card is arXiv:math/0608208v6 (dated Jan. 10, 2012; 42 pp.). The arXiv record names arXiv's non-exclusive distribution license (arXiv:math/0608208), every other right reserved.

Read status: claims checked, for the statements the bibliography makes; it proves nothing. The abstract, §1 and §2 (pp. 1--2) and entry 51 (p. 18) were read on the page images.

Bears on.

• #1135: the problem's map $f$ is the bibliography's $T$ on the positive integers, and its question is the bibliography's $3x+1$ Conjecture (p. 1), which the abstract records as unsolved; see the conjecture page. Entry 51 (p. 18), the annotation of Krasikov and Lagarias (2003), Acta Arith. 109, reports the lower bound $\pi_a(x)>x^{.841}$ for all sufficiently large $x$ when $a\not\equiv0\pmod3$, where $\pi_a(x)$ counts the integers $n\le x$ having some iterate $T^{(k)}(n)=a$; the bibliography reports this bound and does not prove it.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.