least quadratic residue under GRH: an EXPLICIT bound
$begingroup$
Let $m$ be a positive integer and $chi$ a primitive character mod $m$. Let $x$ be such that $chi(p)ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need the best possible bound, but I need a good quality bound which is totally explicit in all parameters.
A related question: what is an explicit lower bound for $L(1,chi)$ under GRH?
nt.number-theory analytic-number-theory l-functions
edited 2 hours ago
Alexey Ustinov
7,00945980
asked 3 hours ago
Yuri BiluYuri Bilu
212
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add a comment |
$begingroup$
Let $m$ be a positive integer and $chi$ a primitive character mod $m$. Let $x$ be such that $chi(p)ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need the best possible bound, but I need a good quality bound which is totally explicit in all parameters.
A related question: what is an explicit lower bound for $L(1,chi)$ under GRH?
nt.number-theory analytic-number-theory l-functions
edited 2 hours ago
Alexey Ustinov
7,00945980
asked 3 hours ago
Yuri BiluYuri Bilu
212
New contributor
Yuri Bilu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let $m$ be a positive integer and $chi$ a primitive character mod $m$. Let $x$ be such that $chi(p)ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need the best possible bound, but I need a good quality bound which is totally explicit in all parameters.
A related question: what is an explicit lower bound for $L(1,chi)$ under GRH?
nt.number-theory analytic-number-theory l-functions
edited 2 hours ago
Alexey Ustinov
7,00945980
asked 3 hours ago
Yuri BiluYuri Bilu
212
New contributor
Yuri Bilu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Let $m$ be a positive integer and $chi$ a primitive character mod $m$. Let $x$ be such that $chi(p)ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need the best possible bound, but I need a good quality bound which is totally explicit in all parameters.
A related question: what is an explicit lower bound for $L(1,chi)$ under GRH?
nt.number-theory analytic-number-theory l-functions
nt.number-theory analytic-number-theory l-functions
edited 2 hours ago
Alexey Ustinov
7,00945980
asked 3 hours ago
Yuri BiluYuri Bilu
212
New contributor
Yuri Bilu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Alexey Ustinov
7,00945980
asked 3 hours ago
Yuri BiluYuri Bilu
212
New contributor
Yuri Bilu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Alexey Ustinov
7,00945980
edited 2 hours ago
Alexey Ustinov
7,00945980
edited 2 hours ago
Alexey Ustinov
7,00945980
7,00945980
asked 3 hours ago
Yuri BiluYuri Bilu
212
New contributor
Yuri Bilu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
Yuri BiluYuri Bilu
212
asked 3 hours ago
Yuri BiluYuri Bilu
212
212
New contributor
Yuri Bilu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Yuri Bilu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Yuri Bilu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
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See the work of Lamzouri, Li, and Soundararajan (I link the arXiv version; the paper appeared in Math. Comp.). Assuming that $chi$ is a primitive quadratic character (as the title suggests) then Theorem 1.4 of that paper gives an explicit bound on the least prime quadratic residue on GRH. (Indeed that theorem gives an explicit bound on the least prime in any coset of a subgroup of $({Bbb Z}/q{Bbb Z})^times$.) Theorem 1.5 there gives explicit upper and lower bounds for $|L(1,chi)|$ for any primitive character $chi pmod q$ (not necessarily quadratic).
answered 2 hours ago
LuciaLucia
34.7k5150176
$endgroup$
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
1 hour ago
add a comment |
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1 Answer
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1 Answer
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$begingroup$
See the work of Lamzouri, Li, and Soundararajan (I link the arXiv version; the paper appeared in Math. Comp.). Assuming that $chi$ is a primitive quadratic character (as the title suggests) then Theorem 1.4 of that paper gives an explicit bound on the least prime quadratic residue on GRH. (Indeed that theorem gives an explicit bound on the least prime in any coset of a subgroup of $({Bbb Z}/q{Bbb Z})^times$.) Theorem 1.5 there gives explicit upper and lower bounds for $|L(1,chi)|$ for any primitive character $chi pmod q$ (not necessarily quadratic).
answered 2 hours ago
LuciaLucia
34.7k5150176
$endgroup$
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
1 hour ago
add a comment |
$begingroup$
See the work of Lamzouri, Li, and Soundararajan (I link the arXiv version; the paper appeared in Math. Comp.). Assuming that $chi$ is a primitive quadratic character (as the title suggests) then Theorem 1.4 of that paper gives an explicit bound on the least prime quadratic residue on GRH. (Indeed that theorem gives an explicit bound on the least prime in any coset of a subgroup of $({Bbb Z}/q{Bbb Z})^times$.) Theorem 1.5 there gives explicit upper and lower bounds for $|L(1,chi)|$ for any primitive character $chi pmod q$ (not necessarily quadratic).
answered 2 hours ago
LuciaLucia
34.7k5150176
$endgroup$
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
1 hour ago
add a comment |
$begingroup$
See the work of Lamzouri, Li, and Soundararajan (I link the arXiv version; the paper appeared in Math. Comp.). Assuming that $chi$ is a primitive quadratic character (as the title suggests) then Theorem 1.4 of that paper gives an explicit bound on the least prime quadratic residue on GRH. (Indeed that theorem gives an explicit bound on the least prime in any coset of a subgroup of $({Bbb Z}/q{Bbb Z})^times$.) Theorem 1.5 there gives explicit upper and lower bounds for $|L(1,chi)|$ for any primitive character $chi pmod q$ (not necessarily quadratic).
answered 2 hours ago
LuciaLucia
34.7k5150176
$endgroup$
See the work of Lamzouri, Li, and Soundararajan (I link the arXiv version; the paper appeared in Math. Comp.). Assuming that $chi$ is a primitive quadratic character (as the title suggests) then Theorem 1.4 of that paper gives an explicit bound on the least prime quadratic residue on GRH. (Indeed that theorem gives an explicit bound on the least prime in any coset of a subgroup of $({Bbb Z}/q{Bbb Z})^times$.) Theorem 1.5 there gives explicit upper and lower bounds for $|L(1,chi)|$ for any primitive character $chi pmod q$ (not necessarily quadratic).
answered 2 hours ago
LuciaLucia
34.7k5150176
answered 2 hours ago
LuciaLucia
34.7k5150176
answered 2 hours ago
LuciaLucia
34.7k5150176
answered 2 hours ago
LuciaLucia
34.7k5150176
34.7k5150176
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
1 hour ago
add a comment |
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
1 hour ago
1
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
1 hour ago
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
1 hour ago
add a comment |
Yuri Bilu is a new contributor. Be nice, and check out our Code of Conduct.
Yuri Bilu is a new contributor. Be nice, and check out our Code of Conduct.
Yuri Bilu is a new contributor. Be nice, and check out our Code of Conduct.
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