Chebyshev's bias-conjecture and the Riemann Hypothesis












4














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







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    4














    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



























      4












      4








      4







      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















      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













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      edited 2 days ago









      Martin Sleziak

      2,92032028




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      asked 2 days ago









      Dimitris ValianatosDimitris Valianatos

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          16














          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



















          • 1




            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.
            – KConrad
            2 days ago












          • Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
            – Greg Martin
            2 days ago










          • @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
            – kodlu
            yesterday








          • 1




            @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.
            – KConrad
            yesterday






          • 2




            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/…
            – KConrad
            yesterday





















          9














          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





















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            16














            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



















            • 1




              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.
              – KConrad
              2 days ago












            • Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
              – Greg Martin
              2 days ago










            • @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
              – kodlu
              yesterday








            • 1




              @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.
              – KConrad
              yesterday






            • 2




              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/…
              – KConrad
              yesterday


















            16














            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



















            • 1




              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.
              – KConrad
              2 days ago












            • Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
              – Greg Martin
              2 days ago










            • @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
              – kodlu
              yesterday








            • 1




              @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.
              – KConrad
              yesterday






            • 2




              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/…
              – KConrad
              yesterday
















            16












            16








            16






            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














            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 11 hours ago









            Martin Sleziak

            2,92032028




            2,92032028










            answered 2 days ago









            Greg MartinGreg Martin

            8,32813559




            8,32813559








            • 1




              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.
              – KConrad
              2 days ago












            • Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
              – Greg Martin
              2 days ago










            • @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
              – kodlu
              yesterday








            • 1




              @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.
              – KConrad
              yesterday






            • 2




              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/…
              – KConrad
              yesterday
















            • 1




              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.
              – KConrad
              2 days ago












            • Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
              – Greg Martin
              2 days ago










            • @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
              – kodlu
              yesterday








            • 1




              @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.
              – KConrad
              yesterday






            • 2




              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/…
              – KConrad
              yesterday










            1




            1




            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.
            – KConrad
            2 days ago






            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.
            – KConrad
            2 days ago














            Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
            – Greg Martin
            2 days ago




            Sorry, brain fart on my end—it was indeed Landau. Thanks @KConrad
            – Greg Martin
            2 days ago












            @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
            – kodlu
            yesterday






            @GregMartin (sorry for the dumb question) does $e^{-u}$ in your answer stand for $exp{-2pi i u}$?
            – kodlu
            yesterday






            1




            1




            @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.
            – KConrad
            yesterday




            @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.
            – KConrad
            yesterday




            2




            2




            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/…
            – KConrad
            yesterday






            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/…
            – KConrad
            yesterday













            9














            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


























              9














              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
























                9












                9








                9






                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












                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 2 days ago









                kodlukodlu

                3,60921727




                3,60921727






























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