Lichtman 2022 primes arithmetic progressions large moduli shifted

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corollary_1_2: For all sufficiently large x there are at least x^0.3389 Carmichael numbers up to x.

corollary_1_3: The integers m for which phi(n) = m has at least m^0.7156 solutions form an infinite sequence whose consecutive terms satisfy log m_(i+1)/log m_i -> 1.

theorem_1_1: For fixed nonzero a and every beta > 15/(32 sqrt e) = 0.2843..., at least x/(log x)^C primes p in (x, 2x] have every prime factor of p - a at most x^beta.

theorem_1_4: A mean value theorem for primes in arithmetic progressions to moduli qrst with divisor-bounded weights in each variable, under QR < x^(1/2+eps), QS^2 < x^(1/2-2eps) and S^2 < R < x^(1/32-eps), reaching moduli up to x^(17/32-eps).

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Jared Duker Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022). The copy read for this card is arXiv:2211.09641v1 (14 November 2022; the print is dated November 4, 2022), 27 pages. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2211.09641), every other right reserved.

Lichtman proves (Theorem 1.1, p. 1) that for fixed nonzero $a$ and every $\beta>15/(32\sqrt e)=0.2843\ldots$ there is $C\ge1$ such that $\gg x/(\log x)^C$ primes $p\in(x,2x]$ have $P^+(p-a)\le x^\beta$, where $P^+$ is the largest prime factor. This refines the exponent $0.2961$ of Baker and Harman (1998); the table on p. 1 traces the earlier exponents back to Erdős (1935), who showed that some $\delta>0$ has infinitely many primes with $P^+(p-1)\le p^{1-\delta}$. The paper frames the result under Erdős's conjecture that $P^+(p-a)\le p^\varepsilon$ infinitely often for every $\varepsilon>0$.

The main technical input is Theorem 1.4 (p. 2), a mean value theorem for primes in arithmetic progressions to moduli of the product form $qrst$: for fixed nonzero $a$, $\varepsilon>0$ and $Q,R,S$ with $QR<x^{1/2+\varepsilon}$, $QS^2<x^{1/2-2\varepsilon}$ and $S^2<R<x^{1/32-\varepsilon}$, and complex weights on $q\le Q$, $r\le R$, $s\le S$, $t\le S$ bounded by $\tau(\cdot)^{B_0}$, the weighted sum over $(qrst,a)=1$ of $\pi(x;qrst,a)-\pi(x)/\varphi(qrst)$ is $\ll_{a,\varepsilon,A}x/(\log x)^A$ for every $A>0$. The paper notes that this may handle moduli up to $x^{17/32-\varepsilon}$, beyond Maynard's $x^{11/21-\varepsilon}$ and the $x^{29/56}$ of Bombieri, Friedlander and Iwaniec (1986). Section 3 (pp. 4--6) deduces Theorem 1.1 from Theorem 1.4; Sections 5 to 12 (pp. 7--26) prove Theorem 1.4 from four propositions modelled on Maynard's.

Two consequences are recorded on p. 2: at least $x^{0.3389}$ Carmichael numbers up to $x$ for large $x$ (Corollary 1.2), from Theorem 1.1 and Harman's form of the Alford--Granville--Pomerance bound, against $0.3333\ldots$ from Baker and Harman's exponent; and the integers $m$ with at least $m^{0.7156}$ solutions of $\varphi(n)=m$ form an infinite sequence with $\log m_{i+1}/\log m_i\to1$ (Corollary 1.3), by the method of Erdős and Pomerance.

Source: https://arxiv.org/abs/2211.09641.

Read status. Claims checked: Theorems 1.1 and 1.4, display (1.3) and Corollaries 1.2 and 1.3 were read clause by clause on the page images of pp. 1--2. The deduction of Theorem 1.1 in Section 3 (pp. 4--6) was read for its structure only; the proof of Theorem 1.4 (pp. 7--26) was not checked. Nothing here is independently reviewed.

Result pages: Theorem 1.1 (p. 1), Corollary 1.2 (p. 2), Corollary 1.3 (p. 2) and Theorem 1.4 (p. 2).

Bears on.

• #821: the problem asks whether, for every $\epsilon>0$, infinitely many $n$ have more than $n^{1-\epsilon}$ solutions of $\varphi(m)=n$. Corollary 1.3 gives infinitely many $n$ with at least $n^{0.7156}$ solutions, which answers the question for every $\epsilon>0.2844$ and says nothing for smaller $\epsilon$. The corpus's claim page Lichtman 2022 extends this to $\epsilon=0.2844$ through Theorem 1.1, a step the paper does not state.
• #1057: the problem asks whether the number $C(x)$ of Carmichael numbers up to $x$ is $x^{1-o(1)}$. Corollary 1.2 gives $C(x)\ge x^{0.3389}$ for large $x$, a lower bound with a fixed exponent that does not reach the question.

Theorems 1.1 and 1.4 bear on these problems only through the two corollaries.

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