Least quadratic residue under GRH: an explicit bound The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)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
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)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
29.1k486140
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 Apr 8 at 14:07
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 Apr 8 at 14:07
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 Apr 8 at 14:07
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 Apr 8 at 14:07
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 Apr 8 at 14:07
YCor
29.1k486140
edited Apr 8 at 14:07
YCor
29.1k486140
edited Apr 8 at 14:07
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
1
active
oldest
votes
$begingroup$
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
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
Apr 8 at 9:30
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Yuri Bilu is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327447%2fleast-quadratic-residue-under-grh-an-explicit-bound%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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
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
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
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
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
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 $(Bbb Z/qBbb Z)^times$.) 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
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
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 |
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.
Yuri Bilu is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327447%2fleast-quadratic-residue-under-grh-an-explicit-bound%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
