I am studying Arithmetic Large Sieve from following notes of Zeev Rudnick:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html
I have questions in lecture 14 here: http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html precisely in the proof 2.1. ( Page 6 and Page 7)
I am not able to deduce on how should I use lemma 3.3 to write this: $$\sum_{ a_2 mod p_2}^* | \mathcal L(a_1/p_1 + a_2/ p_2) |^2 = \sum_{ a_2 mod p_2}^{*} \left| \sum_{ M < n \leq N+M} a_n e( n a_2/q_2) \right|^2 \geq \frac{ \omega(p_2) } { p_2 - \omega(p_2)} |\mathcal L ( a_1/ q_1)|^2 $$ and how did author deduced that ( Also by lemma 3.3) $$ \sum_{a_1 (mod p_1) }^{*} | L (a_1/ q_1)|^2 \geq \frac{ \omega(p_1) } { p_1 - \omega( p_1)} |L(0)|^2 $$
On page 7, I have a problem in deducing
$$ |\mathcal L(0)|^2 \sum_{q \leq z , q-square-free} \frac{ \omega(q)} { q \prod_{p| q} ( 1- \omega(p)/p)} \leq \sum_{q\leq z} \sum_{ a(mod q)}^{*} |\mathcal L(a/q)|^2 \leq |\mathcal L(0)| (N+z^2) $$
When I used the statement of Theorem 3.1 on the inequality on RHS I am not getting $|\mathcal L(0)|$, rest of the inequality I am able to deduce.
Kindly help me with it.
Lemma 3.3 actually holds in a much more general case:
$$ |\mathcal L(\alpha)|^2{\omega(p)\over p-\omega(p)}\le\mathop{\sum\nolimits^*}_{a\pmod p}\left|\mathcal L\left(\frac ap+\alpha\right)\right|^2, $$
and this can be proven by replacing $a_n$ with $a_ne(n\alpha)$ in the proof provided in the lecture note the OP mentioned.
Applying this inequality repetitively (the CRT argument in the lecture note you mentioned) will give
$$ |\mathcal L(0)|^2\sum_{\substack{q\le z\\q\text{ squarefree}}}\prod_{p|d}{p\over p-\omega(p)}\le\sum_{q\le z}\mathop{\sum\nolimits^*}_{a\pmod q}\left|\mathcal L\left(\frac aq\right)\right|^2. $$
Applying the standard large sieve inequality on the right hand side gives you the desired result.