I'm reading Wright's A simple proof of a theorem of Landau in which the core argument is a proof by induction and I find myself stuck on a major point. I must be misunderstanding notation or something else, because what I understand is clearly wrong.
For the base case, he writes:
But, for $k=1,$ (6) is equivalent to (2).
where (2) is a form of the Prime Number Theorem, $$ \vartheta(x)=\sum_{p\le x}\log p\sim x $$
and (6) is $$ \phi_k(x)=o\left\{(\log\log x)^{k-1}\right\} $$
The relevant definitions: $$ \begin{align} \phi_k(x)&=\vartheta_k(x)-kx\Omega_{k-1}(x)\\ \vartheta_k(x)&=\sum_{p_1\cdots p_k\le x}\log(p_1\cdots p_k)\\ \Omega_0(x)&=1\\ \Omega_k(x)&=\sum_{p_1\cdots p_k\le x}\frac{1}{p_1\cdots p_k}\text{ for }k>0\\ \end{align} $$ (where both $\vartheta_k$ and $\Omega_k$ count their sums with multiplicity) and so $$ \phi_1(x)=\vartheta(x)-x $$ but $|\psi(x)-x|\ne o(\sqrt x\log\log\log x)$ and $\psi(x)-\vartheta(x)=O(\sqrt x)$ hence $|\vartheta(x)-x|\ne o(\sqrt x\log\log\log x),$ and so $\phi_1(x)=\vartheta(x)-x\ne o(1)$ as required. (In any case it can't be $o(1)$ since there are arbitrarily long gaps in the primes.) What am I missing?
It looks like a typo to me, and that equation $(6)$ should read $$\phi_k(x)=o\left(x\left(\log \log x\right)^{k-1}\right).$$
Knowing how the author defines $\Omega_{k-1}(x)$ for other values of $k$ would help confirm this.