Least quadratic residue under GRH: an explicit boundexplicit lower bounds on $|L(1,chi)|$Explicit bound on...
Least quadratic residue under GRH: an explicit bound
explicit lower bounds on $|L(1,chi)|$Explicit bound on $sum_{Nmathfrak p leq x}chi(mathfrak p)ln(Nmathfrak p)$Explicit bounds for exceptional zeros and/or $L(1,chi)$ for real $chi$Effective bound of $L(1,chi)$Property of Dirichlet characterOn a sequence of L-functions having same zeros in critical strip and GRHQuestion about the term $sum_{ rho} frac{X^{rho}}{rho}$ in the explicit formula of $sum_{n leq X} Lambda(n) chi(n)$Questions about the exceptional zeros of Dirichlet $L$-functionsPrime character sumsExplicit Version of the Burgess Theorem
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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 12 hours ago
YCor
29k486140
asked yesterday
Yuri BiluYuri Bilu
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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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$begingroup$
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 12 hours ago
YCor
29k486140
asked yesterday
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 12 hours ago
YCor
29k486140
asked yesterday
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 12 hours ago
YCor
29k486140
asked yesterday
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 12 hours ago
YCor
29k486140
asked yesterday
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 12 hours ago
YCor
29k486140
edited 12 hours ago
YCor
29k486140
edited 12 hours ago
YCor
29k486140
29k486140
asked yesterday
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 yesterday
Yuri BiluYuri Bilu
835
asked yesterday
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 yesterday
LuciaLucia
34.9k5151177
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1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
yesterday
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
17 hours 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 yesterday
LuciaLucia
34.9k5151177
$endgroup$
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
yesterday
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
17 hours 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 yesterday
LuciaLucia
34.9k5151177
$endgroup$
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
yesterday
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
17 hours 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 yesterday
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 yesterday
LuciaLucia
34.9k5151177
answered yesterday
LuciaLucia
34.9k5151177
answered yesterday
LuciaLucia
34.9k5151177
answered yesterday
LuciaLucia
34.9k5151177
34.9k5151177
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
yesterday
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
17 hours ago
add a comment |
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
yesterday
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
17 hours ago
1
1
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
yesterday
$begingroup$
Lucia, many thanks! This is exactly what I am looking for!
$endgroup$
– Yuri Bilu
yesterday
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
17 hours 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
17 hours ago
add a comment |
Yuri Bilu is a new contributor. Be nice, and check out our Code of Conduct.
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