A question related to inequality in Selberg Sieve

Question

Consider these images related to Theory of selberg Sieve , taken from notes on Sieve theory Of Prof. Rudnick. http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html

Note: Earlier it was asked on Mathoverflow but due to no response I am asking here and will delete from there. Image 1:

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Image 2 :

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In last Line of 2nd Image I am unable to deduce how author wrote $z^2$ ie how from (4) $O(\sum_{d_1 ,d_2}|\lambda_{d_1} | |\lambda_{d_2}|)$ is proved to be $\leq z^2$.

It is given that d$\leq z$ and $\lambda_d =0$ for d>z but I am unable to find a relation between $\lambda_d$ and z.

Can you please help with that.

Answer 1

By manipulating the main term with Cauchy-Schwarz inequality, it is possible to show that $|\lambda_d|\le1$ (Have a look at Nathanson's GTM164 Additive Number Theory). Thus the latter sum will be $\le z^2$. Hope this addresses your concern!