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 days ago
YCor
29.1k486140
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
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Yuri Bilu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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 days ago
YCor
29.1k486140
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
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 days ago
YCor
29.1k486140
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
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 days ago
YCor
29.1k486140
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
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 days ago
YCor
29.1k486140
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
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 days ago
YCor
29.1k486140
edited 2 days ago
YCor
29.1k486140
edited 2 days ago
YCor
29.1k486140
29.1k486140
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
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 Apr 8 at 1:21
Yuri BiluYuri Bilu
835
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
835
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 Apr 8 at 2:05
LuciaLucia
34.9k5151177
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1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
Apr 8 at 2:37
2
$begingroup$
@YuriBilu: If you like Lucia's answer, please accept it officially (so that it turns green). Thanks! (And welcome to MO!)
$endgroup$
– GH from MO
2 days 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 Apr 8 at 2:05
LuciaLucia
34.9k5151177
$endgroup$
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
Apr 8 at 2:37
2
$begingroup$
@YuriBilu: If you like Lucia's answer, please accept it officially (so that it turns green). Thanks! (And welcome to MO!)
$endgroup$
– GH from MO
2 days 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 Apr 8 at 2:05
LuciaLucia
34.9k5151177
$endgroup$
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
Apr 8 at 2:37
2
$begingroup$
@YuriBilu: If you like Lucia's answer, please accept it officially (so that it turns green). Thanks! (And welcome to MO!)
$endgroup$
– GH from MO
2 days 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 Apr 8 at 2:05
LuciaLucia
34.9k5151177
$endgroup$
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 Apr 8 at 2:05
LuciaLucia
34.9k5151177
answered Apr 8 at 2:05
LuciaLucia
34.9k5151177
answered Apr 8 at 2:05
LuciaLucia
34.9k5151177
answered Apr 8 at 2:05
LuciaLucia
34.9k5151177
34.9k5151177
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
Apr 8 at 2:37
2
$begingroup$
@YuriBilu: If you like Lucia's answer, please accept it officially (so that it turns green). Thanks! (And welcome to MO!)
$endgroup$
– GH from MO
2 days ago
add a comment |
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
Apr 8 at 2:37
2
$begingroup$
@YuriBilu: If you like Lucia's answer, please accept it officially (so that it turns green). Thanks! (And welcome to MO!)
$endgroup$
– GH from MO
2 days ago
1
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
Apr 8 at 2:37
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
Apr 8 at 2:37
2
2
$begingroup$
@YuriBilu: If you like Lucia's answer, please accept it officially (so that it turns green). Thanks! (And welcome to MO!)
$endgroup$
– GH from MO
2 days ago
$begingroup$
@YuriBilu: If you like Lucia's answer, please accept it officially (so that it turns green). Thanks! (And welcome to MO!)
$endgroup$
– GH from MO
2 days 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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