Joint distribution in residue classes of polynomial-like multiplicative functions

Two problems on the distribution of Carmichael's lambda function

“…Actually, [16, Proposition 2.1] estimates the frequency with which a general “polynomial‐like” multiplicative function f is coprime to a given integer q . Taking $$f=\varphi$$ gives Proposition 2.1, after observing that the conditions $$(\lambda (n),q)=1$$ and $$(\varphi (n),q)=1$$ are equivalent.…”

“…(For any finite abelian group, the exponent and order share the same set of prime factors.) We note that in [16], the exponent of $$\log _2{(3q)}$$ is given as O (1), but inspecting the proof reveals that this exponent can be taken as 2 for $$f=\varphi$$.…”

“…We also set $$$$\begin{equation*} y: = \exp (\sqrt {\log {x}}). \end{equation*}$$$$Following [16, section 3] (choosing $$\delta :=1$$ in the notation of that paper), a positive integer $$n\leqslant x$$ is called convenient if $$P_J(n) > y$$ and none of the primes $$P_1(n), P_{2}(n), \dots , P_J(n)$$ have squares dividing n . That is, n is convenient when n admits a decomposition $$$$\begin{equation} n = mP_J \cdots P_1, \quad \text{with}\quad L_m:=\max \lbrace P^{+}(m),y\rbrace < P_J < \cdots < P_1.$$…”

“…Write $$N = N(x,q)$$ for the total count of $$n\leqslant x$$ with $$(\lambda (n),q)=1$$. The following lemma, proved as Lemma 3.1 in [16], shows that this count does not change very much if we restrict to convenient n . Lemma The number of inconvenient $$n\leqslant x$$ with $$(\lambda (n),q)=1$$ is $$o(N)$$.…”

“…As λ is not multiplicative, we must take a different tack. In recent work with Singha Roy [16], we proposed an alternative method of proving UD and WUD theorems. This was used to show (among other things) that for $$f=\varphi$$, the relation (1.1) holds uniformly for $$q\leqslant (\log {x})^{A}$$ with $$\gcd (q,6)=1$$.…”