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







share|cite|improve this question





























    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













      share|cite|improve this question




      share|cite|improve this question








      edited 2 days ago









      Martin Sleziak

      2,92032028




      2,92032028










      asked 2 days ago









      Dimitris ValianatosDimitris Valianatos

      628412




      628412






















          2 Answers
          2






          active

          oldest

          votes


















          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





















            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "504"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f320102%2fchebyshevs-bias-conjecture-and-the-riemann-hypothesis%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            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






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to MathOverflow!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.





                    Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                    Please pay close attention to the following guidance:


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f320102%2fchebyshevs-bias-conjecture-and-the-riemann-hypothesis%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    -

                    Popular posts from this blog

                    Mario Kart Wii

                    Image
                    Please consider supporting The Cutting Room Floor on Patreon. Thanks for all your support! Mario Kart Wii From The Cutting Room Floor Jump to: navigation, search Mario Kart Wii Developer : Nintendo Publisher : Nintendo Platform : Wii Released in JP : April 10, 2008 Released in US : April 27, 2008 Released in EU : April 11, 2008 Released in AU : April 24, 2008 Released in KR : April 30, 2009 This game has unused animations. This game has unused areas. This game has unused code. This game has unused objects. This game has unused modes / minigames. This game has unused graphics. This game has unused models. This game has unused items. This game has unused music. This game has regional differences. This game has a notes page This game has a prerelease article Mario Kart Wii is the sixth game in the series (not counting Arcade editions), which intro...
                    Read more

                    The Binding of Isaac: Rebirth/Afterbirth

                    Image
                    Please consider supporting The Cutting Room Floor on Patreon. Thanks for all your support! The Binding of Isaac: Rebirth/Afterbirth From The Cutting Room Floor < The Binding of Isaac: Rebirth Jump to: navigation, search This is a sub-page of The Binding of Isaac: Rebirth . Afterbirth is the first expansion pack for Rebirth and features a variety of improvements. Contents 1 Unused Graphics 1.1 Old Man Keeper 1.2 Skinless Hush 1.3 Miscellaneous Unused Hush Sprites 1.4 Generosity achievement 1.5 Afterbirth+ Room Editor 1.6 Bearded Ultra Greed 1.7 Early Ultra Greed 1.8 Unused Brownie Expression 1.9 Frail's Spinning Head 1.10 Mom's Dead Hand Alt 2 Unused Audio 2.1 Unused Dead Womb Layer 2.2 Unused Hush Battle Outro 2.3 The Stain Grunt Unused Graphics Old Man Keeper Image file: gfxuimain me...
                    Read more

                    What does “Dominus providebit” mean?

                    Image
                    5 Does "Dominus providebit" mean "The Lord will provide" or "The Lord provides"? I once had a bit of an argument with a guy who studied Latin over that. It's an inscription that appears on the rim of some old Swiss coins that I have. meaning latin-to-english-translation grammar-identification share | improve this question edited Jan 18 at 15:27 V2Blast 105 4 asked Jan 17 at 22:49 Donald K. Donald K. 26 1 ...
                    Read more