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I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski. It is found on top of page 200. In bottom of page 199, the authors prove that $$ |S_f(N)|^2 \leq N + \frac{2N^2}{q} + 4(N+q)\log(q). $$ They claim that using the last inequality and the trivial bound $ |S_f(N)| \leq N $, (8.11) follows, namely $$ |S_f(N)| \leq2\frac{N}{\sqrt{q}} + \sqrt{q}\log(q). $$

I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski. It is found on top of page 200. In bottom of page 199, the authors prove that $$ |S_f(N)|^2 \leq N + \frac{2N^2}{q} + 4(N+q)\log(q). $$ They claim that using the last inequality and the trivial bound $ |S_f(N)| \leq N $, (8.11) follows, namely $$ |S_f(N)| \leq2\frac{N}{\sqrt{q}} + \sqrt{q}\log(q). $$ where $ S_f(N) = \sum_{n=1}^N e^{\alpha n^2 + \beta n}, $ where $ \alpha $ and $ \beta $ are constants. also $ \left| 2\alpha - \frac{a}{q} \right| \leq \frac{1}{2Nq}, $ where $(a, q) = 1$ and $1 \leq q \leq 2N$.

I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski, found at the end of page 199. The inequality in question is: $$ |S_f(N)|^2 \leq N + \frac{2N^2}{q} + 4(N+q)\log(q). $$ Using the trivial bound $ |S_f(N)| \leq N $, they state that this inequality implies (8.11): $$ |S_f(N)| \leq2\frac{N}{\sqrt{q}} + \sqrt{q}\log(q). $$

I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski. It is found on top of page 200. In bottom of page 199, the authors prove that $$ |S_f(N)|^2 \leq N + \frac{2N^2}{q} + 4(N+q)\log(q). $$ They claim that using the last inequality and the trivial bound $ |S_f(N)| \leq N $, (8.11) follows, namely $$ |S_f(N)| \leq2\frac{N}{\sqrt{q}} + \sqrt{q}\log(q). $$

I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski, found at the end of page 199. The inequality in question is: $$ |S_f(N)|^2 \leq N + \frac{2N^2}{q} + 4(N+q)\log(q). $$ Using the trivial bound $ |S_f(N)| \leq N $, they state that this inequality implies (8.11): $$ |S_f(N)| \leq2\sqrt{q} + \sqrt{q}\log(q). $$

I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski, found at the end of page 199. The inequality in question is: $$ |S_f(N)|^2 \leq N + \frac{2N^2}{q} + 4(N+q)\log(q). $$ Using the trivial bound $ |S_f(N)| \leq N $, they state that this inequality implies (8.11): $$ |S_f(N)| \leq2\frac{N}{\sqrt{q}} + \sqrt{q}\log(q). $$