Least quadratic residue under GRH: an explicit bound Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?explicit lower bounds on $|L(1,chi)|$Explicit bound on $sum_Nmathfrak p leq xchi(mathfrak p)ln(Nmathfrak p)$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 $sum_ rho fracX^rhorho$ 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
Least quadratic residue under GRH: an explicit bound
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?explicit lower bounds on $|L(1,chi)|$Explicit bound on $sum_Nmathfrak p leq xchi(mathfrak p)ln(Nmathfrak p)$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 $sum_ rho fracX^rhorho$ 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
$begingroup$
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 Apr 8 at 14:07
YCor
29.1k487141
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
$endgroup$
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 Apr 8 at 14:07
YCor
29.1k487141
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
$endgroup$
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 Apr 8 at 14:07
YCor
29.1k487141
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
$endgroup$
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 Apr 8 at 14:07
YCor
29.1k487141
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
edited Apr 8 at 14:07
YCor
29.1k487141
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
edited Apr 8 at 14:07
YCor
29.1k487141
edited Apr 8 at 14:07
YCor
29.1k487141
edited Apr 8 at 14:07
YCor
29.1k487141
29.1k487141
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
asked Apr 8 at 1:21
Yuri BiluYuri Bilu
835
835
add a comment |
add a comment |
1 Answer
1
active
oldest
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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 $(Bbb Z/qBbb Z)^times$.) 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
35.2k5151179
$endgroup$
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
Apr 8 at 9:30
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 $(Bbb Z/qBbb Z)^times$.) 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
35.2k5151179
$endgroup$
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
Apr 8 at 9:30
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 $(Bbb Z/qBbb Z)^times$.) 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
35.2k5151179
$endgroup$
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
Apr 8 at 9:30
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 $(Bbb Z/qBbb Z)^times$.) 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
35.2k5151179
$endgroup$
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 $(Bbb Z/qBbb Z)^times$.) 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
35.2k5151179
answered Apr 8 at 2:05
LuciaLucia
35.2k5151179
answered Apr 8 at 2:05
LuciaLucia
35.2k5151179
answered Apr 8 at 2:05
LuciaLucia
35.2k5151179
35.2k5151179
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
Apr 8 at 9:30
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
Apr 8 at 9:30
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
Apr 8 at 9:30
$begingroup$
@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
Apr 8 at 9:30
add a comment |
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