least quadratic residue under GRH: an EXPLICIT bound












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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







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    1












    $begingroup$


    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







    share|cite|improve this question









    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.







    $endgroup$















      1












      1








      1





      $begingroup$


      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







      share|cite|improve this question









      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






      share|cite|improve this question









      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.











      share|cite|improve this question









      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.









      share|cite|improve this question




      share|cite|improve this question








      edited 2 hours ago









      Alexey Ustinov

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      7,00945980






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      asked 2 hours ago









      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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          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).






          share|cite|improve this answer









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            Lucia, many thanks! This is exactly what I am looking for!
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            – Yuri Bilu
            1 hour ago












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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).






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Lucia, many thanks! This is exactly what I am looking for!
            $endgroup$
            – Yuri Bilu
            1 hour ago
















          3












          $begingroup$

          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).






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Lucia, many thanks! This is exactly what I am looking for!
            $endgroup$
            – Yuri Bilu
            1 hour ago














          3












          3








          3





          $begingroup$

          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).






          share|cite|improve this answer









          $endgroup$



          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).







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 1 hour 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














          • 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










          Yuri Bilu is a new contributor. Be nice, and check out our Code of Conduct.










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