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I am currently reading Iwaniec-Kowaleski's book on Analytic Number Theory.

My question is on page 431. Here is what it says:

For any $chi mod q$, we have the twisted series $K(s,chi)=sum_{n=1}^infty left( sum_{d|n} lambda_d right) left( sum_{d|n} theta_b right) chi(n) n^{-s}.$

Here, we have $lambda(d)= mu(d) min left(1, frac{log(z/d)}{log(z/w)} right)$ for $1 leq d leq z$ where $1<w<z$ and we set $lambda_d=0$ if $d>z$.

Thus, we have $sum_{d|n} lambda_d=1$ if $n=1$ and $sum_{d|n} lambda_d=0$ if $1<n leq w.$

Furthermore, we have
$theta_b = frac{mu(b) b}{G phi(b)} sum_{ab leq y atop (a,bq)=1} frac{mu^2 (a)}{phi(a)}$ and $G$ is the normalization
factor such that $theta_1=1.$

Now, we can see that $K(s,chi)$ factors into $L(s,chi)M(s,chi)$, where $M(s,chi)=sum_m left(sum_{[b,d]} sum_{=m} lambda_d theta_b right) chi(m) m^{-s}$.

Now, if we take only a partial sum of $K(s,chi)$, we can define
$K_x(s,chi)=sum_{n=1}^x left( sum_{d|n} lambda_d right) left( sum_{d|n} theta_b right) chi(n) n^{-s}$ with $x=(qT)^{23}.$
Let $m=[b,d] leq bd leq yz $ and $chi not= chi_0.$

Then, why does this imply $left|sum_{n>x/m} chi(n) n^{-s} right| leq 2q |s| (m/x)^{sigma}$ where $s=sigma +it$. I do not understand where the $2q|s|$ comes from.

Assuming this, the book states
$|K(s,chi)-K_x(s,chi)| leq 2q|s|yz x^{-sigma}$. I do not understand where the yz term comes from.

Sorry for the length of this post. I wanted to give as much information as possible.

Showing $left|sum_{n>x/m} chi(n) n^{-s} right| leq 2q |s| (m/x)^{sigma}$.

I will assume that $q > 1$ and that $sigma geq 1/2$. Then
$$sum_{n geq N} frac{chi(n)}{n^s} = sum_{n geq N} S_chi(n) left( frac{1}{n^s} - frac{1}{(n+1)^s}right) tag{1},$$
where $S_chi(n) = sum_{N leq m leq n} chi(m)$. This is an application of partial summation.

A trivial bound on $S_chi(n)$ is $lvert S_chi(n) rvert leq q$. (It is here that I use that $q > 1$). In fact, this is a very weak bound, but that's ok.

I note that for $mathrm{Re}(s) > 0$, and $0 < a < b$, we have
$$ left lvert frac{1}{a^s} - frac{1}{b^s}right rvert leq frac{lvert s rvert}{sigma} left( frac{1}{a^sigma} - frac{1}{b^sigma}right) leq frac{lvert s rvert(b - a)}{a^{sigma + 1}}.$$
For a quick proof, look at
$$ int_a^b frac{dx}{x^{s+1}} = frac{1}{s} left( frac{1}{a^s} - frac{1}{b^s}right),$$
and thus
$$ left lvert frac{1}{a^s} - frac{1}{b^s}right rvert leq lvert s rvert int_a^b frac{dx}{lvert x^{sigma+1} rvert},$$
giving the first inequality. For the second inequality, one can apply the mean value theorem to $1/x^{sigma}$, or a trivial integral estimate.

In this case, this gives that
$$ leftlvert frac{1}{n^s} - frac{1}{(n+1)^s}rightrvert leq frac{lvert s rvert}{n^{sigma + 1}}.$$
Applying to $(1)$ shows that
$$ sum_{n geq N} frac{chi(n)}{n^s} leq sum_{n geq N} q frac{lvert s rvert}{n^{sigma + 1}} leq 2 q lvert s rvert N^{-sigma}.$$
In your case, you have $N = x/m$, and this proves the claim. $diamondsuit$

$|K(s,chi)-K_x(s,chi)| leq 2q|s|yz x^{-sigma}$

I didn't work out the complete details, but I did work out part of the argument. In short, I didn't want to dive into any of the details concerning $x,y,z$ or $theta_b, lambda_d$, and so I use only trivial bounds for those parts.

From IK, we have the following properties:

• $x = (qT)^{23}$


• $y = (qT)^2$


• $z = (qT)^8$


• $w = (qt)^7$

And also $lambda_d = 0$ if $d > z$, and $theta_b = 0$ if $b > y$. Further, $lvert lambda_d rvert leq 1$ and $lvert theta_b rvert leq 1$. (IK has a whole section devoted to $theta_b$ and the Selberg sieve, and restates some relevant bounds here for more precise bounds than I give below).

Then one can write
$$ K_x(s, chi) = sum_{m geq 1} left( sum_{[b, d] = m} sum lambda_d theta_b right) frac{chi(m)}{m^s} sum_{n leq x/m} frac{chi(n)}{n^s}$$
(in the same factoring sort-of argument that shows that $K(s, chi) = M(s, chi) L(s, chi)$). Thus
$$ lvert K(s, chi) - K_x(s, chi) rvert leq
left lvert sum_{m geq 1} sum_{[b,d] = m} lambda_d theta_b frac{chi(m)}{m^s}right rvert
left lvert sum_{n geq x/m} chi(n)n^{-s}right rvert. tag{2}$$

The latter part of $(2)$ is exactly what was considered above, and as
$m leq bd leq yz$, we have that
$$ left lvert sum_{n > x/m} chi(n) n^{-s} right rvert leq 2 q lvert s rvert (yz)^sigma x^{-sigma}.$$

For the first sum of $(2)$, let us naively and lossfully note that
$$ sum_{[b,d] = n} lambda_d theta_b leq tau(n),$$
the number of divisors of $n$.
Then the first $lvert cdot rvert$ part can then be bounded above by
$$ sum_{n leq yz} tau(n) n^{-sigma} leq yz log(yz) left(1+ frac{(yz)^{- sigma}}{sigma}right) leq 2(yz)^{1 - sigma} log(yz).$$

Putting these together, we find that $(2)$ is bounded above by
$$2(yz)^{1 - sigma} log(yz) 2 q lvert s rvert (yz)^sigma x^{-sigma}
= 4 q lvert s rvert yz log(yz) x^{-sigma}.$$
This is slightly weaker than the stated result in IK, but I think this describes how to get the sort of result as stated.