[proofplan] We apply the locally staged multiplicative large sieve for primitive Dirichlet characters with the same interval, moduli, and coefficient sequence. That theorem gives the sharper explicit factor $N-1+Q^2$. Since $N-1+Q^2\le N+Q^2$, the exact bound immediately implies the stated Vinogradov-notation estimate. [/proofplan] [step:Invoke the primitive-character large sieve] The locally staged Multiplicative Large Sieve for Primitive Dirichlet Characters [quotetheorem:TEMP-26] applies to the integers $M$, $N$, and $Q$ and to the finite coefficient sequence $(a_n)_{M<n\le M+N}$. It gives \begin{align*} \sum_{q \leq Q} \frac{q}{\varphi(q)} \sum_{\chi \bmod q}^{*} \left|\sum_{M < n \leq M+N} a_n \chi(n)\right|^2 \leq (N-1+Q^2) \sum_{M < n \leq M+N} |a_n|^2. \end{align*} [guided] The theorem we need has exactly the same left-hand side as the primitive-character multiplicative large sieve already staged in these notes. The hypotheses also match: the coefficients are supported on the interval $M<n\le M+N$, and the outer sum is over moduli $q\le Q$ with primitive characters modulo $q$. Therefore we may use that theorem directly. [/guided] [/step] [step:Pass from the explicit constant to Vinogradov notation] Because $N\in\mathbb N$, we have \begin{align*} N-1+Q^2 \le N+Q^2. \end{align*} Substituting this into the preceding inequality gives \begin{align*} \sum_{q \leq Q} \frac{q}{\varphi(q)} \sum_{\chi \bmod q}^{*} \left|\sum_{M < n \leq M+N} a_n \chi(n)\right|^2 \leq (N+Q^2) \sum_{M < n \leq M+N} |a_n|^2. \end{align*} This is the asserted estimate with an absolute implicit constant. [guided] The displayed estimate is stronger than the statement. Vinogradov notation $\lesssim$ with an absolute implicit constant allows multiplication of the right-hand side by a fixed positive constant. Here no loss is needed, because $N-1+Q^2$ is already at most $N+Q^2$. [/guided] [/step]