Infinitely many Carmichael numbers in arithmetic progressions (Q2854223)

A Carmichael number \(m\in\mathbb N\) is a composite integer such that \(m\mid (a^m-a)\) for every \(a\in\mathbb Z\); by Fermat's little theorem this always holds when \(m\) is a prime. \textit{W. H. Alford} et al. [Ann. Math. (2) 139, No. 3, 703--722 (1994; Zbl 0816.11005)] established the expected result that there are infinitely many Carmichael numbers. This led to other conjectures, in particular one stating that there are infinitely many Carmichael numbers \(m\equiv a\pmod M\) whenever \(M\in\mathbb N\) and \((a,m)=1\). In the case when \(a\) is a quadratic residue \(\pmod M\) this was proved by \textit{K. Matomäki} [J. Aust. Math. Soc. 94, No. 2, 268--275 (2013; Zbl 1368.11106)] who obtained a lower bound \(\gg x^{1/5}\) for the number of such \(m\leq x\). However her proof does not work for quadratic nonresidues \(\pmod M\).NEWLINENEWLINE The present author is able to prove the conjecture completely, and he obtains a lower bound \(\ll X^{K/(\log\log\log X)^2}\) for the number of Carmichael numbers \(m\equiv a\pmod M\) up to \(X\) for any \(M\in\mathbb N\) and \((a,m)=1\), where the constant \(K\) depends on \(M\). His proof is built on those in the two papers cited above but introduces extra requirements. These necessitate a larger \(x\) (relative to \(M\) and another parameter \(L'\)) than that in previous papers, and then \(X\) is taken to be a power of \(x\). His introduction explains very clearly the similarities to and differences from previous related proofs, and the technical steps that follow are well set out.