Chebyshev's bias-conjecture and the Riemann Hypothesis












1














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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    1














    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



























      1












      1








      1







      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






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      edited 10 hours ago









      Martin Sleziak

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      asked 15 hours ago









      Dimitris Valianatos

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          12














          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
            8 hours ago












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



















          6














          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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            12














            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
              8 hours ago












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
















            12














            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
              8 hours ago












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














            12












            12








            12






            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 3 hours ago

























            answered 10 hours ago









            Greg Martin

            8,28813559




            8,28813559








            • 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
              8 hours ago












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














            • 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
              8 hours ago












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








            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
            8 hours 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
            8 hours ago














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




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











            6














            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


























              6














              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
























                6












                6








                6






                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 14 hours ago









                kodlu

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