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










10












$begingroup$


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











$endgroup$
















    10












    $begingroup$


    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











    $endgroup$














      10












      10








      10





      $begingroup$


      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











      $endgroup$




      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















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Apr 8 at 14:07









      YCor

      29.1k487141




      29.1k487141










      asked Apr 8 at 1:21









      Yuri BiluYuri Bilu

      835




      835




















          1 Answer
          1






          active

          oldest

          votes


















          21












          $begingroup$

          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).






          share|cite|improve this answer









          $endgroup$








          • 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












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          1 Answer
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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          21












          $begingroup$

          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).






          share|cite|improve this answer









          $endgroup$








          • 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
















          21












          $begingroup$

          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).






          share|cite|improve this answer









          $endgroup$








          • 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














          21












          21








          21





          $begingroup$

          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).






          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 $(Bbb Z/qBbb Z)^times$.) Theorem 1.5 there gives explicit upper and lower bounds for $|L(1,chi)|$ for any primitive character $chi pmod q$ (not necessarily quadratic).







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          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













          • 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


















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