An observation on Non-Trivial Zeros of Riemann Zeta Function.












1












$begingroup$


I observed this property in month of July this year but unable to design a mathematical proof or mathematical way to state my observation. I need help to state this property.



We know that for certain values of 'b' in s = 1/2 + ib , ζ(s) = 0.



When I observed most of the values of 'b' here



I found that most of the primes are related to these values of 'b' in there square forms like:



    Property:    [b] = p^2        {where 'p' is a prime number and [b] is the Box Function}


example: b = 841.0363...



    so,         [841.0363...] = 29^2


below 200 there are only 6 primes which are not following the this above property.



I am searching a formula or a program to find out all those primes which follow above property but till now I didn't got any solution. I am also not good in programming please help me in this problem.










share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    I observed this property in month of July this year but unable to design a mathematical proof or mathematical way to state my observation. I need help to state this property.



    We know that for certain values of 'b' in s = 1/2 + ib , ζ(s) = 0.



    When I observed most of the values of 'b' here



    I found that most of the primes are related to these values of 'b' in there square forms like:



        Property:    [b] = p^2        {where 'p' is a prime number and [b] is the Box Function}


    example: b = 841.0363...



        so,         [841.0363...] = 29^2


    below 200 there are only 6 primes which are not following the this above property.



    I am searching a formula or a program to find out all those primes which follow above property but till now I didn't got any solution. I am also not good in programming please help me in this problem.










    share|cite|improve this question









    $endgroup$















      1












      1








      1


      1



      $begingroup$


      I observed this property in month of July this year but unable to design a mathematical proof or mathematical way to state my observation. I need help to state this property.



      We know that for certain values of 'b' in s = 1/2 + ib , ζ(s) = 0.



      When I observed most of the values of 'b' here



      I found that most of the primes are related to these values of 'b' in there square forms like:



          Property:    [b] = p^2        {where 'p' is a prime number and [b] is the Box Function}


      example: b = 841.0363...



          so,         [841.0363...] = 29^2


      below 200 there are only 6 primes which are not following the this above property.



      I am searching a formula or a program to find out all those primes which follow above property but till now I didn't got any solution. I am also not good in programming please help me in this problem.










      share|cite|improve this question









      $endgroup$




      I observed this property in month of July this year but unable to design a mathematical proof or mathematical way to state my observation. I need help to state this property.



      We know that for certain values of 'b' in s = 1/2 + ib , ζ(s) = 0.



      When I observed most of the values of 'b' here



      I found that most of the primes are related to these values of 'b' in there square forms like:



          Property:    [b] = p^2        {where 'p' is a prime number and [b] is the Box Function}


      example: b = 841.0363...



          so,         [841.0363...] = 29^2


      below 200 there are only 6 primes which are not following the this above property.



      I am searching a formula or a program to find out all those primes which follow above property but till now I didn't got any solution. I am also not good in programming please help me in this problem.







      number-theory prime-numbers riemann-zeta zeta-functions






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 17 '18 at 13:50









      DynamoDynamo

      104517




      104517






















          1 Answer
          1






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          3












          $begingroup$

          The $n$th zeta zero has $b(n) sim 2pi n/W(n/e)$, where $W$ is the Lambert function. The Lambert function grows as roughly the logarithm of its argument, so we see that the zeta zeroes get denser as $n$ increases, and at $n approx 10^4$, their imaginary parts should start hitting every integer. This turns out to be the case, as the last square of a prime it misses is $103^2 = 10609$.



          We can also estimate the probability of each square of a prime not having a zeta zero near it. The density of zeta zeroes is $W(n/e)/(2pi)$, so we expect each square of a prime to not be near a zeta zero with probability $1 - W(p^2/e)/(2pi)$. Summing this over all primes where this value is positive, we get an expected count of $7.7$, which is reasonably close to the actual value of 6.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for your answer but are you sure that after n=10^4 their imaginary part start hitting every integer. If yes then please tell me where I can see its proof.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 16:02












          • $begingroup$
            @AdarshKumar My statement is probably a little too strong, but it is known that there is some $n$ after which $[b(n)]$ hits every integer. It is also known how to count the number of zeros with imaginary part less than $b$, with formula given here. Keeping only the dominant terms and inverting gives my estimate in the answer. It is possible there are integers missed that are significantly larger than $10^4$, but prime squares have very low density at large $n$, making them unlikely to be counterexamples anyways.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:09












          • $begingroup$
            so it means that after some n which you say that will surely exist after which b(n) hits every integer than it is also possible all primes whose square is greater than n also exist . So my property is true till infinite primes. ??
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:15










          • $begingroup$
            Yes, this property will hold for all sufficiently large primes. Though since this is true because it holds for all sufficiently large integers, of which primes are a subset, I'm not sure how much this tells us about primes.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:18










          • $begingroup$
            ok thanks a lot to giving your time to solve my problem.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:21











          Your Answer





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          3












          $begingroup$

          The $n$th zeta zero has $b(n) sim 2pi n/W(n/e)$, where $W$ is the Lambert function. The Lambert function grows as roughly the logarithm of its argument, so we see that the zeta zeroes get denser as $n$ increases, and at $n approx 10^4$, their imaginary parts should start hitting every integer. This turns out to be the case, as the last square of a prime it misses is $103^2 = 10609$.



          We can also estimate the probability of each square of a prime not having a zeta zero near it. The density of zeta zeroes is $W(n/e)/(2pi)$, so we expect each square of a prime to not be near a zeta zero with probability $1 - W(p^2/e)/(2pi)$. Summing this over all primes where this value is positive, we get an expected count of $7.7$, which is reasonably close to the actual value of 6.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for your answer but are you sure that after n=10^4 their imaginary part start hitting every integer. If yes then please tell me where I can see its proof.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 16:02












          • $begingroup$
            @AdarshKumar My statement is probably a little too strong, but it is known that there is some $n$ after which $[b(n)]$ hits every integer. It is also known how to count the number of zeros with imaginary part less than $b$, with formula given here. Keeping only the dominant terms and inverting gives my estimate in the answer. It is possible there are integers missed that are significantly larger than $10^4$, but prime squares have very low density at large $n$, making them unlikely to be counterexamples anyways.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:09












          • $begingroup$
            so it means that after some n which you say that will surely exist after which b(n) hits every integer than it is also possible all primes whose square is greater than n also exist . So my property is true till infinite primes. ??
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:15










          • $begingroup$
            Yes, this property will hold for all sufficiently large primes. Though since this is true because it holds for all sufficiently large integers, of which primes are a subset, I'm not sure how much this tells us about primes.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:18










          • $begingroup$
            ok thanks a lot to giving your time to solve my problem.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:21
















          3












          $begingroup$

          The $n$th zeta zero has $b(n) sim 2pi n/W(n/e)$, where $W$ is the Lambert function. The Lambert function grows as roughly the logarithm of its argument, so we see that the zeta zeroes get denser as $n$ increases, and at $n approx 10^4$, their imaginary parts should start hitting every integer. This turns out to be the case, as the last square of a prime it misses is $103^2 = 10609$.



          We can also estimate the probability of each square of a prime not having a zeta zero near it. The density of zeta zeroes is $W(n/e)/(2pi)$, so we expect each square of a prime to not be near a zeta zero with probability $1 - W(p^2/e)/(2pi)$. Summing this over all primes where this value is positive, we get an expected count of $7.7$, which is reasonably close to the actual value of 6.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks for your answer but are you sure that after n=10^4 their imaginary part start hitting every integer. If yes then please tell me where I can see its proof.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 16:02












          • $begingroup$
            @AdarshKumar My statement is probably a little too strong, but it is known that there is some $n$ after which $[b(n)]$ hits every integer. It is also known how to count the number of zeros with imaginary part less than $b$, with formula given here. Keeping only the dominant terms and inverting gives my estimate in the answer. It is possible there are integers missed that are significantly larger than $10^4$, but prime squares have very low density at large $n$, making them unlikely to be counterexamples anyways.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:09












          • $begingroup$
            so it means that after some n which you say that will surely exist after which b(n) hits every integer than it is also possible all primes whose square is greater than n also exist . So my property is true till infinite primes. ??
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:15










          • $begingroup$
            Yes, this property will hold for all sufficiently large primes. Though since this is true because it holds for all sufficiently large integers, of which primes are a subset, I'm not sure how much this tells us about primes.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:18










          • $begingroup$
            ok thanks a lot to giving your time to solve my problem.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:21














          3












          3








          3





          $begingroup$

          The $n$th zeta zero has $b(n) sim 2pi n/W(n/e)$, where $W$ is the Lambert function. The Lambert function grows as roughly the logarithm of its argument, so we see that the zeta zeroes get denser as $n$ increases, and at $n approx 10^4$, their imaginary parts should start hitting every integer. This turns out to be the case, as the last square of a prime it misses is $103^2 = 10609$.



          We can also estimate the probability of each square of a prime not having a zeta zero near it. The density of zeta zeroes is $W(n/e)/(2pi)$, so we expect each square of a prime to not be near a zeta zero with probability $1 - W(p^2/e)/(2pi)$. Summing this over all primes where this value is positive, we get an expected count of $7.7$, which is reasonably close to the actual value of 6.






          share|cite|improve this answer









          $endgroup$



          The $n$th zeta zero has $b(n) sim 2pi n/W(n/e)$, where $W$ is the Lambert function. The Lambert function grows as roughly the logarithm of its argument, so we see that the zeta zeroes get denser as $n$ increases, and at $n approx 10^4$, their imaginary parts should start hitting every integer. This turns out to be the case, as the last square of a prime it misses is $103^2 = 10609$.



          We can also estimate the probability of each square of a prime not having a zeta zero near it. The density of zeta zeroes is $W(n/e)/(2pi)$, so we expect each square of a prime to not be near a zeta zero with probability $1 - W(p^2/e)/(2pi)$. Summing this over all primes where this value is positive, we get an expected count of $7.7$, which is reasonably close to the actual value of 6.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 17 '18 at 15:43









          eyeballfrogeyeballfrog

          6,228629




          6,228629












          • $begingroup$
            Thanks for your answer but are you sure that after n=10^4 their imaginary part start hitting every integer. If yes then please tell me where I can see its proof.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 16:02












          • $begingroup$
            @AdarshKumar My statement is probably a little too strong, but it is known that there is some $n$ after which $[b(n)]$ hits every integer. It is also known how to count the number of zeros with imaginary part less than $b$, with formula given here. Keeping only the dominant terms and inverting gives my estimate in the answer. It is possible there are integers missed that are significantly larger than $10^4$, but prime squares have very low density at large $n$, making them unlikely to be counterexamples anyways.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:09












          • $begingroup$
            so it means that after some n which you say that will surely exist after which b(n) hits every integer than it is also possible all primes whose square is greater than n also exist . So my property is true till infinite primes. ??
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:15










          • $begingroup$
            Yes, this property will hold for all sufficiently large primes. Though since this is true because it holds for all sufficiently large integers, of which primes are a subset, I'm not sure how much this tells us about primes.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:18










          • $begingroup$
            ok thanks a lot to giving your time to solve my problem.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:21


















          • $begingroup$
            Thanks for your answer but are you sure that after n=10^4 their imaginary part start hitting every integer. If yes then please tell me where I can see its proof.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 16:02












          • $begingroup$
            @AdarshKumar My statement is probably a little too strong, but it is known that there is some $n$ after which $[b(n)]$ hits every integer. It is also known how to count the number of zeros with imaginary part less than $b$, with formula given here. Keeping only the dominant terms and inverting gives my estimate in the answer. It is possible there are integers missed that are significantly larger than $10^4$, but prime squares have very low density at large $n$, making them unlikely to be counterexamples anyways.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:09












          • $begingroup$
            so it means that after some n which you say that will surely exist after which b(n) hits every integer than it is also possible all primes whose square is greater than n also exist . So my property is true till infinite primes. ??
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:15










          • $begingroup$
            Yes, this property will hold for all sufficiently large primes. Though since this is true because it holds for all sufficiently large integers, of which primes are a subset, I'm not sure how much this tells us about primes.
            $endgroup$
            – eyeballfrog
            Dec 17 '18 at 17:18










          • $begingroup$
            ok thanks a lot to giving your time to solve my problem.
            $endgroup$
            – Dynamo
            Dec 17 '18 at 17:21
















          $begingroup$
          Thanks for your answer but are you sure that after n=10^4 their imaginary part start hitting every integer. If yes then please tell me where I can see its proof.
          $endgroup$
          – Dynamo
          Dec 17 '18 at 16:02






          $begingroup$
          Thanks for your answer but are you sure that after n=10^4 their imaginary part start hitting every integer. If yes then please tell me where I can see its proof.
          $endgroup$
          – Dynamo
          Dec 17 '18 at 16:02














          $begingroup$
          @AdarshKumar My statement is probably a little too strong, but it is known that there is some $n$ after which $[b(n)]$ hits every integer. It is also known how to count the number of zeros with imaginary part less than $b$, with formula given here. Keeping only the dominant terms and inverting gives my estimate in the answer. It is possible there are integers missed that are significantly larger than $10^4$, but prime squares have very low density at large $n$, making them unlikely to be counterexamples anyways.
          $endgroup$
          – eyeballfrog
          Dec 17 '18 at 17:09






          $begingroup$
          @AdarshKumar My statement is probably a little too strong, but it is known that there is some $n$ after which $[b(n)]$ hits every integer. It is also known how to count the number of zeros with imaginary part less than $b$, with formula given here. Keeping only the dominant terms and inverting gives my estimate in the answer. It is possible there are integers missed that are significantly larger than $10^4$, but prime squares have very low density at large $n$, making them unlikely to be counterexamples anyways.
          $endgroup$
          – eyeballfrog
          Dec 17 '18 at 17:09














          $begingroup$
          so it means that after some n which you say that will surely exist after which b(n) hits every integer than it is also possible all primes whose square is greater than n also exist . So my property is true till infinite primes. ??
          $endgroup$
          – Dynamo
          Dec 17 '18 at 17:15




          $begingroup$
          so it means that after some n which you say that will surely exist after which b(n) hits every integer than it is also possible all primes whose square is greater than n also exist . So my property is true till infinite primes. ??
          $endgroup$
          – Dynamo
          Dec 17 '18 at 17:15












          $begingroup$
          Yes, this property will hold for all sufficiently large primes. Though since this is true because it holds for all sufficiently large integers, of which primes are a subset, I'm not sure how much this tells us about primes.
          $endgroup$
          – eyeballfrog
          Dec 17 '18 at 17:18




          $begingroup$
          Yes, this property will hold for all sufficiently large primes. Though since this is true because it holds for all sufficiently large integers, of which primes are a subset, I'm not sure how much this tells us about primes.
          $endgroup$
          – eyeballfrog
          Dec 17 '18 at 17:18












          $begingroup$
          ok thanks a lot to giving your time to solve my problem.
          $endgroup$
          – Dynamo
          Dec 17 '18 at 17:21




          $begingroup$
          ok thanks a lot to giving your time to solve my problem.
          $endgroup$
          – Dynamo
          Dec 17 '18 at 17:21


















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