Chebyshev's bias-conjecture and the Riemann Hypothesis












4












$begingroup$


Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?






nt.number-theory analytic-number-theory prime-numbers riemann-hypothesis







share|cite|improve this question











$endgroup$

















    4












    $begingroup$


    Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?






    nt.number-theory analytic-number-theory prime-numbers riemann-hypothesis







    share|cite|improve this question











    $endgroup$















      4












      4








      4


      1



      $begingroup$


      Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?






      nt.number-theory analytic-number-theory prime-numbers riemann-hypothesis







      share|cite|improve this question











      $endgroup$




      Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?






      nt.number-theory analytic-number-theory prime-numbers riemann-hypothesis




      nt.number-theory analytic-number-theory prime-numbers riemann-hypothesis






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 5 at 2:12









      Martin Sleziak

      3,09032231




      3,09032231










      asked Jan 4 at 20:59









      Dimitris ValianatosDimitris Valianatos

      628412




      628412






















          2 Answers
          2






          active

          oldest

          votes


















          17












          $begingroup$

          Chebyshev made the following assertion, which is interpreted as saying that there are more primes of the form $4n+3$ than of the form $4n+1$:
          $$
          lim_{xtoinfty} sum_{pge3} (-1)^{(p-1)/2} e^{-p/x} = -infty.
          $$
          It was proved by Hardy/Littlewood and Landau that this assertion is equivalent to the generalized Riemann hypothesis for the Dirichlet $L$-function
          $$
          L(s,chi_{-4}) = 1 - 3^{-s} + 5^{-s} - 7^{-s} + 9^{-s} - 11^{-s} + cdots
          $$
          corresponding to the nonprincipal character (mod 4).




          • G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), no. 1, 119–196

          • E. Landau, Über einige ältere Vermutungen und Behauptungen in der Primzahltheorie, Math. Z. 1 (1918), 1–24






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            What was the paper of Ingham? I only knew that Landau in 1918 had shown the equivalence (Hardy and Littlewood did GRH for the character mod 4 implies Chebyshev's conjecture). See Math. Zeitschrift 1 (1918), pp. 1-24 and 213-219.
            $endgroup$
            – KConrad
            Jan 5 at 4:17












          • $begingroup$
            Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
            $endgroup$
            – Greg Martin
            Jan 5 at 8:48










          • $begingroup$
            @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
            $endgroup$
            – kodlu
            Jan 6 at 1:35








          • 1




            $begingroup$
            @kodlu what Greg writes is correct. For each fixed $x > 0$, $e^{-p/x} rightarrow 0$ as $p rightarrow infty$ through the primes; the series converges for each $x$. Using something like $e^{-2pi i p/x}$ could make each term of the series of magnitude 1 and it would not converge.
            $endgroup$
            – KConrad
            Jan 6 at 16:47






          • 2




            $begingroup$
            Chebyshev actually wrote the series with all signs changed, so using $(-1)^{(p+1)/2}$, and his variable was $c = 1/x$ tending to $0$, so his conjecture was $sum_{p > 2} (-1)^{(p+1)/2}e^{-pc} rightarrow infty$ as $c rightarrow 0^+$ with the sum running over the odd primes. See what Chebyshev wrote in Google books: books.google.com/…
            $endgroup$
            – KConrad
            Jan 6 at 16:51





















          12












          $begingroup$

          Chebyshev's bias is consistent with the Riemann Hypothesis. I like Terry Tao's explanation in his blog:




          [..] the bias is small [..] and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function ${Lambda}$ modulo ${q}$ with the zeroes of the ${L}$-functions with period ${q}$. This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function ${Lambda}$ is quite unbiased modulo ${q}.$ The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo ${q}.$ (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias.




          See also Rubinstein and Sarnak MR review here.






          share|cite|improve this answer









          $endgroup$














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            17












            $begingroup$

            Chebyshev made the following assertion, which is interpreted as saying that there are more primes of the form $4n+3$ than of the form $4n+1$:
            $$
            lim_{xtoinfty} sum_{pge3} (-1)^{(p-1)/2} e^{-p/x} = -infty.
            $$
            It was proved by Hardy/Littlewood and Landau that this assertion is equivalent to the generalized Riemann hypothesis for the Dirichlet $L$-function
            $$
            L(s,chi_{-4}) = 1 - 3^{-s} + 5^{-s} - 7^{-s} + 9^{-s} - 11^{-s} + cdots
            $$
            corresponding to the nonprincipal character (mod 4).




            • G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), no. 1, 119–196

            • E. Landau, Über einige ältere Vermutungen und Behauptungen in der Primzahltheorie, Math. Z. 1 (1918), 1–24






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              What was the paper of Ingham? I only knew that Landau in 1918 had shown the equivalence (Hardy and Littlewood did GRH for the character mod 4 implies Chebyshev's conjecture). See Math. Zeitschrift 1 (1918), pp. 1-24 and 213-219.
              $endgroup$
              – KConrad
              Jan 5 at 4:17












            • $begingroup$
              Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
              $endgroup$
              – Greg Martin
              Jan 5 at 8:48










            • $begingroup$
              @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
              $endgroup$
              – kodlu
              Jan 6 at 1:35








            • 1




              $begingroup$
              @kodlu what Greg writes is correct. For each fixed $x > 0$, $e^{-p/x} rightarrow 0$ as $p rightarrow infty$ through the primes; the series converges for each $x$. Using something like $e^{-2pi i p/x}$ could make each term of the series of magnitude 1 and it would not converge.
              $endgroup$
              – KConrad
              Jan 6 at 16:47






            • 2




              $begingroup$
              Chebyshev actually wrote the series with all signs changed, so using $(-1)^{(p+1)/2}$, and his variable was $c = 1/x$ tending to $0$, so his conjecture was $sum_{p > 2} (-1)^{(p+1)/2}e^{-pc} rightarrow infty$ as $c rightarrow 0^+$ with the sum running over the odd primes. See what Chebyshev wrote in Google books: books.google.com/…
              $endgroup$
              – KConrad
              Jan 6 at 16:51


















            17












            $begingroup$

            Chebyshev made the following assertion, which is interpreted as saying that there are more primes of the form $4n+3$ than of the form $4n+1$:
            $$
            lim_{xtoinfty} sum_{pge3} (-1)^{(p-1)/2} e^{-p/x} = -infty.
            $$
            It was proved by Hardy/Littlewood and Landau that this assertion is equivalent to the generalized Riemann hypothesis for the Dirichlet $L$-function
            $$
            L(s,chi_{-4}) = 1 - 3^{-s} + 5^{-s} - 7^{-s} + 9^{-s} - 11^{-s} + cdots
            $$
            corresponding to the nonprincipal character (mod 4).




            • G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), no. 1, 119–196

            • E. Landau, Über einige ältere Vermutungen und Behauptungen in der Primzahltheorie, Math. Z. 1 (1918), 1–24






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              What was the paper of Ingham? I only knew that Landau in 1918 had shown the equivalence (Hardy and Littlewood did GRH for the character mod 4 implies Chebyshev's conjecture). See Math. Zeitschrift 1 (1918), pp. 1-24 and 213-219.
              $endgroup$
              – KConrad
              Jan 5 at 4:17












            • $begingroup$
              Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
              $endgroup$
              – Greg Martin
              Jan 5 at 8:48










            • $begingroup$
              @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
              $endgroup$
              – kodlu
              Jan 6 at 1:35








            • 1




              $begingroup$
              @kodlu what Greg writes is correct. For each fixed $x > 0$, $e^{-p/x} rightarrow 0$ as $p rightarrow infty$ through the primes; the series converges for each $x$. Using something like $e^{-2pi i p/x}$ could make each term of the series of magnitude 1 and it would not converge.
              $endgroup$
              – KConrad
              Jan 6 at 16:47






            • 2




              $begingroup$
              Chebyshev actually wrote the series with all signs changed, so using $(-1)^{(p+1)/2}$, and his variable was $c = 1/x$ tending to $0$, so his conjecture was $sum_{p > 2} (-1)^{(p+1)/2}e^{-pc} rightarrow infty$ as $c rightarrow 0^+$ with the sum running over the odd primes. See what Chebyshev wrote in Google books: books.google.com/…
              $endgroup$
              – KConrad
              Jan 6 at 16:51
















            17












            17








            17





            $begingroup$

            Chebyshev made the following assertion, which is interpreted as saying that there are more primes of the form $4n+3$ than of the form $4n+1$:
            $$
            lim_{xtoinfty} sum_{pge3} (-1)^{(p-1)/2} e^{-p/x} = -infty.
            $$
            It was proved by Hardy/Littlewood and Landau that this assertion is equivalent to the generalized Riemann hypothesis for the Dirichlet $L$-function
            $$
            L(s,chi_{-4}) = 1 - 3^{-s} + 5^{-s} - 7^{-s} + 9^{-s} - 11^{-s} + cdots
            $$
            corresponding to the nonprincipal character (mod 4).




            • G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), no. 1, 119–196

            • E. Landau, Über einige ältere Vermutungen und Behauptungen in der Primzahltheorie, Math. Z. 1 (1918), 1–24






            share|cite|improve this answer











            $endgroup$



            Chebyshev made the following assertion, which is interpreted as saying that there are more primes of the form $4n+3$ than of the form $4n+1$:
            $$
            lim_{xtoinfty} sum_{pge3} (-1)^{(p-1)/2} e^{-p/x} = -infty.
            $$
            It was proved by Hardy/Littlewood and Landau that this assertion is equivalent to the generalized Riemann hypothesis for the Dirichlet $L$-function
            $$
            L(s,chi_{-4}) = 1 - 3^{-s} + 5^{-s} - 7^{-s} + 9^{-s} - 11^{-s} + cdots
            $$
            corresponding to the nonprincipal character (mod 4).




            • G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), no. 1, 119–196

            • E. Landau, Über einige ältere Vermutungen und Behauptungen in der Primzahltheorie, Math. Z. 1 (1918), 1–24







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Jan 7 at 6:37









            Martin Sleziak

            3,09032231




            3,09032231










            answered Jan 5 at 2:08









            Greg MartinGreg Martin

            8,90813761




            8,90813761








            • 1




              $begingroup$
              What was the paper of Ingham? I only knew that Landau in 1918 had shown the equivalence (Hardy and Littlewood did GRH for the character mod 4 implies Chebyshev's conjecture). See Math. Zeitschrift 1 (1918), pp. 1-24 and 213-219.
              $endgroup$
              – KConrad
              Jan 5 at 4:17












            • $begingroup$
              Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
              $endgroup$
              – Greg Martin
              Jan 5 at 8:48










            • $begingroup$
              @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
              $endgroup$
              – kodlu
              Jan 6 at 1:35








            • 1




              $begingroup$
              @kodlu what Greg writes is correct. For each fixed $x > 0$, $e^{-p/x} rightarrow 0$ as $p rightarrow infty$ through the primes; the series converges for each $x$. Using something like $e^{-2pi i p/x}$ could make each term of the series of magnitude 1 and it would not converge.
              $endgroup$
              – KConrad
              Jan 6 at 16:47






            • 2




              $begingroup$
              Chebyshev actually wrote the series with all signs changed, so using $(-1)^{(p+1)/2}$, and his variable was $c = 1/x$ tending to $0$, so his conjecture was $sum_{p > 2} (-1)^{(p+1)/2}e^{-pc} rightarrow infty$ as $c rightarrow 0^+$ with the sum running over the odd primes. See what Chebyshev wrote in Google books: books.google.com/…
              $endgroup$
              – KConrad
              Jan 6 at 16:51
















            • 1




              $begingroup$
              What was the paper of Ingham? I only knew that Landau in 1918 had shown the equivalence (Hardy and Littlewood did GRH for the character mod 4 implies Chebyshev's conjecture). See Math. Zeitschrift 1 (1918), pp. 1-24 and 213-219.
              $endgroup$
              – KConrad
              Jan 5 at 4:17












            • $begingroup$
              Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
              $endgroup$
              – Greg Martin
              Jan 5 at 8:48










            • $begingroup$
              @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
              $endgroup$
              – kodlu
              Jan 6 at 1:35








            • 1




              $begingroup$
              @kodlu what Greg writes is correct. For each fixed $x > 0$, $e^{-p/x} rightarrow 0$ as $p rightarrow infty$ through the primes; the series converges for each $x$. Using something like $e^{-2pi i p/x}$ could make each term of the series of magnitude 1 and it would not converge.
              $endgroup$
              – KConrad
              Jan 6 at 16:47






            • 2




              $begingroup$
              Chebyshev actually wrote the series with all signs changed, so using $(-1)^{(p+1)/2}$, and his variable was $c = 1/x$ tending to $0$, so his conjecture was $sum_{p > 2} (-1)^{(p+1)/2}e^{-pc} rightarrow infty$ as $c rightarrow 0^+$ with the sum running over the odd primes. See what Chebyshev wrote in Google books: books.google.com/…
              $endgroup$
              – KConrad
              Jan 6 at 16:51










            1




            1




            $begingroup$
            What was the paper of Ingham? I only knew that Landau in 1918 had shown the equivalence (Hardy and Littlewood did GRH for the character mod 4 implies Chebyshev's conjecture). See Math. Zeitschrift 1 (1918), pp. 1-24 and 213-219.
            $endgroup$
            – KConrad
            Jan 5 at 4:17






            $begingroup$
            What was the paper of Ingham? I only knew that Landau in 1918 had shown the equivalence (Hardy and Littlewood did GRH for the character mod 4 implies Chebyshev's conjecture). See Math. Zeitschrift 1 (1918), pp. 1-24 and 213-219.
            $endgroup$
            – KConrad
            Jan 5 at 4:17














            $begingroup$
            Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
            $endgroup$
            – Greg Martin
            Jan 5 at 8:48




            $begingroup$
            Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
            $endgroup$
            – Greg Martin
            Jan 5 at 8:48












            $begingroup$
            @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
            $endgroup$
            – kodlu
            Jan 6 at 1:35






            $begingroup$
            @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
            $endgroup$
            – kodlu
            Jan 6 at 1:35






            1




            1




            $begingroup$
            @kodlu what Greg writes is correct. For each fixed $x > 0$, $e^{-p/x} rightarrow 0$ as $p rightarrow infty$ through the primes; the series converges for each $x$. Using something like $e^{-2pi i p/x}$ could make each term of the series of magnitude 1 and it would not converge.
            $endgroup$
            – KConrad
            Jan 6 at 16:47




            $begingroup$
            @kodlu what Greg writes is correct. For each fixed $x > 0$, $e^{-p/x} rightarrow 0$ as $p rightarrow infty$ through the primes; the series converges for each $x$. Using something like $e^{-2pi i p/x}$ could make each term of the series of magnitude 1 and it would not converge.
            $endgroup$
            – KConrad
            Jan 6 at 16:47




            2




            2




            $begingroup$
            Chebyshev actually wrote the series with all signs changed, so using $(-1)^{(p+1)/2}$, and his variable was $c = 1/x$ tending to $0$, so his conjecture was $sum_{p > 2} (-1)^{(p+1)/2}e^{-pc} rightarrow infty$ as $c rightarrow 0^+$ with the sum running over the odd primes. See what Chebyshev wrote in Google books: books.google.com/…
            $endgroup$
            – KConrad
            Jan 6 at 16:51






            $begingroup$
            Chebyshev actually wrote the series with all signs changed, so using $(-1)^{(p+1)/2}$, and his variable was $c = 1/x$ tending to $0$, so his conjecture was $sum_{p > 2} (-1)^{(p+1)/2}e^{-pc} rightarrow infty$ as $c rightarrow 0^+$ with the sum running over the odd primes. See what Chebyshev wrote in Google books: books.google.com/…
            $endgroup$
            – KConrad
            Jan 6 at 16:51













            12












            $begingroup$

            Chebyshev's bias is consistent with the Riemann Hypothesis. I like Terry Tao's explanation in his blog:




            [..] the bias is small [..] and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function ${Lambda}$ modulo ${q}$ with the zeroes of the ${L}$-functions with period ${q}$. This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function ${Lambda}$ is quite unbiased modulo ${q}.$ The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo ${q}.$ (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias.




            See also Rubinstein and Sarnak MR review here.






            share|cite|improve this answer









            $endgroup$


















              12












              $begingroup$

              Chebyshev's bias is consistent with the Riemann Hypothesis. I like Terry Tao's explanation in his blog:




              [..] the bias is small [..] and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function ${Lambda}$ modulo ${q}$ with the zeroes of the ${L}$-functions with period ${q}$. This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function ${Lambda}$ is quite unbiased modulo ${q}.$ The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo ${q}.$ (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias.




              See also Rubinstein and Sarnak MR review here.






              share|cite|improve this answer









              $endgroup$
















                12












                12








                12





                $begingroup$

                Chebyshev's bias is consistent with the Riemann Hypothesis. I like Terry Tao's explanation in his blog:




                [..] the bias is small [..] and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function ${Lambda}$ modulo ${q}$ with the zeroes of the ${L}$-functions with period ${q}$. This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function ${Lambda}$ is quite unbiased modulo ${q}.$ The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo ${q}.$ (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias.




                See also Rubinstein and Sarnak MR review here.






                share|cite|improve this answer









                $endgroup$



                Chebyshev's bias is consistent with the Riemann Hypothesis. I like Terry Tao's explanation in his blog:




                [..] the bias is small [..] and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function ${Lambda}$ modulo ${q}$ with the zeroes of the ${L}$-functions with period ${q}$. This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function ${Lambda}$ is quite unbiased modulo ${q}.$ The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo ${q}.$ (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias.




                See also Rubinstein and Sarnak MR review here.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 4 at 22:02









                kodlukodlu

                4,22721930




                4,22721930






























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