Here is a link on the internet: <https://www.dropbox.com/s/su3uak2a057yrqv/YitangZhang.pdf>


Can someone teach me how to use trivial estimation to reach (6.1) on page 24? Namely, how to impose $(d,P\_0)<D\_1$ and replace $\theta$ by $\Lambda$ with acceptable O-term?


Thanks a lot!



A technical question related to Zhang's result of bounded prime gaps
Let me remind some of notations for readers:
$$\mathcal{L}=\log x,$$
$$k\_0=3.5\times 10^6, \varpi=\frac{1}{1168},$$
$$D\_0=\exp(\mathcal{L}^{\frac{1}{k\_0}}), P\_0=\prod\_{p < D\_0}p,$$
$$D\_1=x^{\varpi},P=\prod\_{p < D\_1}p,$$


$$D=x^{\frac{1}{4}+\varpi}, D\_2 = x^{\frac{1}{2}-\epsilon}$$


Here, the effect of replacing $\theta$ by $\Lambda$ produces an error of 
$$O(x^{\frac{1}{2}}\mathcal{L}^B),$$
for some positive $B$. So, this change is negligible compared to $O(x\mathcal{L}^{-A})$.


For imposing $(d,P\_0)< D\_1$, we claim that the following sum is also negligible compared to $O(x\mathcal{L}^{-A})$:
$$
\sum\_{\substack{{D\_2 < d {<} D^2} \\\ {d|P} \\\ {(d,P\_0)\geq D\_1}}} \sum\_{c\in C\_i(d)} |\Delta(\Lambda,d,c)|
$$
Note that the trivial bound for $|\Delta(\Lambda,d,c)|$ is:
$$
|\Delta(\Lambda,d,c)|=O(\frac{x\mathcal{L}}{d})$$


Since we have $(d,P\_0)\geq D\_1$, the number $w(d)$ of distinct prime divisors of $d$, must satisfy 
$$
w(d )\geq \mathcal{L}^{\varpi(1-\frac{1}{k\_0})}.$$


The sum is bounded by:
$$
x\mathcal{L} \sum\_{\substack{{ d {<} D^2} \\\ {w(d )\geq \mathcal{L}^{\varpi(1-\frac{1}{k\_0})} } }} \frac{\tau\_{k\_0}(d)}{d }$$


Standard argument now applies, and the sum above is bounded by:
$$
\frac{1}{2^{\mathcal{L}^{\varpi(1-\frac{1}{k\_0})}}}\sum\_{d < D^2} \frac{2^{w(d)}\tau\_{k\_0}(d)}{d}=O(\frac{\mathcal{L}^B }{2^{\mathcal{L}^{\varpi(1-\frac{1}{k\_0})}}}),$$
for some positive constant $B$. 


Combining all together, we obtain an upper bound:
$$
\sum\_{\substack{{D\_2 < d {<} D^2} \\\ {d|P} \\\ {(d,P\_0)\geq D\_1}}} \sum\_{c\in C\_i(d)} |\Delta(\Lambda,d,c)| =O(\frac{x\mathcal{L}^{B+1}}{2^{\mathcal{L}^{\varpi(1-\frac{1}{k\_0})}} })$$


Hence, our claim follows. 


