This is definition 1.
Let $A\subseteq\N$ be a set such that $\pi_A(x)=x/\delta+O(x^\alpha)$,
where $\delta\in[1,\infty)$ and $0\le\alpha<1$.
Then we have
$$
M_A(x)=\frac{1}{2\delta}x\log x+
\frac 1 \delta\left(\gamma-\frac{1}{2}\log\frac \delta 2-\frac{1}{2}\right)x
+O_A!\left(x^{\frac{2\alpha+2}{\alpha+3}}\right).
$$
Let $A\subseteq\N$ be a set such that, for some $\delta>0$,
$$
\pi_A(x)=\frac{x}{\delta\log x}\left(1+\frac{1}{\log x}+O!\left({\frac{1}{\log^2 x}}\right)\right)
$$
for all sufficiently large $x$. Then we have, for sufficiently large $x$,
$$
\begin{align*}
M_A(x)
&=
\frac{\log2}{\delta}x-\frac{1}{\delta}\frac{x\log\log x}{\log x}+
\frac{1}{\delta}\left(\gamma-\log\frac{\delta}{4}\right)\frac{x\log\log x}{\log x}\
&\qquad+
\frac{1}{2\delta}\frac{x(\log\log x)^2}{\log^2 x}+O_A!\left(\frac{x\log\log x}{\log^2 x}\right).
\end{align*}
$$