Banks 2018 counting integers smooth totient

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theorem_1_1: For fixed eps > 0, y at least (log log x)^(1+eps) and u = log x/log y tending to infinity, at most x exp(-u(log log u + log log log u + o(1))) integers n up to x have phi(n) free of prime factors exceeding y.

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W. D. Banks, J. B. Friedlander, C. Pomerance, I. E. Shparlinski, Counting integers with a smooth totient. arXiv preprint (2018). arXiv:1809.01214. The copy read for this card is arXiv v1 (submitted 4 September 2018). The paper was later published in Q. J. Math. 70 (2019), no. 4, 1371-1386, doi:10.1093/qmathj/haz026; that version was not compared with the preprint. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1809.01214), every other right reserved.

Write Phi(x,y) for the number of n <= x whose totient phi(n) has no prime factor exceeding y. Theorem 3.1 of the authors' 2004 paper asserted Phi(x,y) <= x/exp((1+o(1))u log log u) for fixed eps > 0, y >= (log log x)^{1+eps} and u = log x/log y -> infinity, but Paul Kinlaw found a flaw in its proof; this note gives a complete proof of a stronger bound, which the abstract calls likely best possible. Theorem 1.1 (p. 1): in the same range, Phi(x,y) <= x exp(-u(log log u + log log log u + o(1))). For problem 1003 Steve Fan, in a comment of 23 February 2026 on the erdosproblems.com forum thread, sketched how retuning the parameters of the Erdos-Pomerance-Sarkozy 1987 proof and using Theorem 1.1 to handle its item (iv) improves their bound x exp(-(log x)^{1/3}) on the number of n <= x with phi(n) = phi(n+1) to x exp(-c_0((log x)(log log x)(log log log x))^{1/3}) for a suitable constant c_0 > 0. The method is analytic counting of integers by the smoothness of their totient.

Source: https://arxiv.org/abs/1809.01214.

Bears on. #1003: the paper does not mention the problem; Theorem 1.1 enters only through Steve Fan's forum sketch above, which bounds the number of n <= x with phi(n) = phi(n+1) from above, is not checked here, and does not decide whether there are infinitely many solutions.

Results. Labels and pages are those of arXiv v1.

• Theorem 1.1 (p. 1): Phi(x,y) <= x exp(-u(log log u + log log log u + o(1))) for fixed eps > 0, y >= (log log x)^{1+eps} and u = log x/log y -> infinity.

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