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Question in Iwaniec-Kowaleski's book : bound for a twisted series
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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.
analytic-number-theory
asked Nov 15 at 17:44
usere5225321
579311
This question has an open bounty worth +50
reputation from usere5225321 ending in 6 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |
up vote
2
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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.
analytic-number-theory
asked Nov 15 at 17:44
usere5225321
579311
This question has an open bounty worth +50
reputation from usere5225321 ending in 6 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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.
analytic-number-theory
asked Nov 15 at 17:44
usere5225321
579311
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.
analytic-number-theory
analytic-number-theory
asked Nov 15 at 17:44
usere5225321
579311
asked Nov 15 at 17:44
usere5225321
579311
asked Nov 15 at 17:44
usere5225321
579311
asked Nov 15 at 17:44
usere5225321
579311
asked Nov 15 at 17:44
usere5225321
579311
579311
This question has an open bounty worth +50
reputation from usere5225321 ending in 6 days.
Looking for an answer drawing from credible and/or official sources.
This question has an open bounty worth +50
reputation from usere5225321 ending in 6 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |
add a comment |
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