Can this function's positivity $forall t gt 5.56..$ be criteria for the Riemann hypothesis?

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Let $Z left( t right) =rm e^i left( -i/2 left( it lnGamma
left( 1/4+i/2t right) -it ln Gamma left( 1/4-i/2t right)
right) -1/2,ln left( pi right) t right) zeta left( 1/2+
it right) $



be the Hardy Z function



then define the integral of the logarithmic derivative of Z



$R(t)=frac 1pi int
!frac frac rm drm dtZ left( t right) Z left( t
right) ,rm dt=frac -1/2,it lnGamma left( 1/4-i/2t right) +1/2,it lnGamma
left( 1/4+i/2t right) +ln left( zeta left( 1/2+it right)
right) -i/2ln left( pi right) tpi $



then let



$G(t)=- left( frac rm d^2rm dt^2R left( t right)
right) ^-1=-8,frac left( zeta left( 1/2+it right)
right) ^2pi Psi left( 1,1/4-i/2t right) left( zeta
left( 1/2+it right) right) ^2-Psi left( 1,1/4+i/2t right)
left( zeta left( 1/2+it right) right) ^2-8,zeta left( 2,
1/2+it right) zeta left( 1/2+it right) +8, left( zeta left(
1,1/2+it right) right) ^2$



where $Psi(1,x)$ is the 1st derivative of the digamma function and $zeta(n,t$) is the n-th derivative of the Riemann zeta function.



The function G(t) has an essential singularity inside the range



$(5.56057204723220 , 5.56180573101957)$



I believe that if there was a root of Zeta off the line then it would cause $G(t)<0$ for some $t>5.56...$ . Therefore, can the Riemann hypothesis be stated as the positivity of $G(t)$ for all real t > $5.56...$ ?



enter image description here




complex-analysis derivatives logarithms riemann-zeta laplacian




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    Let $Z left( t right) =rm e^i left( -i/2 left( it lnGamma
    left( 1/4+i/2t right) -it ln Gamma left( 1/4-i/2t right)
    right) -1/2,ln left( pi right) t right) zeta left( 1/2+
    it right) $



    be the Hardy Z function



    then define the integral of the logarithmic derivative of Z



    $R(t)=frac 1pi int
    !frac frac rm drm dtZ left( t right) Z left( t
    right) ,rm dt=frac -1/2,it lnGamma left( 1/4-i/2t right) +1/2,it lnGamma
    left( 1/4+i/2t right) +ln left( zeta left( 1/2+it right)
    right) -i/2ln left( pi right) tpi $



    then let



    $G(t)=- left( frac rm d^2rm dt^2R left( t right)
    right) ^-1=-8,frac left( zeta left( 1/2+it right)
    right) ^2pi Psi left( 1,1/4-i/2t right) left( zeta
    left( 1/2+it right) right) ^2-Psi left( 1,1/4+i/2t right)
    left( zeta left( 1/2+it right) right) ^2-8,zeta left( 2,
    1/2+it right) zeta left( 1/2+it right) +8, left( zeta left(
    1,1/2+it right) right) ^2$



    where $Psi(1,x)$ is the 1st derivative of the digamma function and $zeta(n,t$) is the n-th derivative of the Riemann zeta function.



    The function G(t) has an essential singularity inside the range



    $(5.56057204723220 , 5.56180573101957)$



    I believe that if there was a root of Zeta off the line then it would cause $G(t)<0$ for some $t>5.56...$ . Therefore, can the Riemann hypothesis be stated as the positivity of $G(t)$ for all real t > $5.56...$ ?



    enter image description here




    complex-analysis derivatives logarithms riemann-zeta laplacian




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      Let $Z left( t right) =rm e^i left( -i/2 left( it lnGamma
      left( 1/4+i/2t right) -it ln Gamma left( 1/4-i/2t right)
      right) -1/2,ln left( pi right) t right) zeta left( 1/2+
      it right) $



      be the Hardy Z function



      then define the integral of the logarithmic derivative of Z



      $R(t)=frac 1pi int
      !frac frac rm drm dtZ left( t right) Z left( t
      right) ,rm dt=frac -1/2,it lnGamma left( 1/4-i/2t right) +1/2,it lnGamma
      left( 1/4+i/2t right) +ln left( zeta left( 1/2+it right)
      right) -i/2ln left( pi right) tpi $



      then let



      $G(t)=- left( frac rm d^2rm dt^2R left( t right)
      right) ^-1=-8,frac left( zeta left( 1/2+it right)
      right) ^2pi Psi left( 1,1/4-i/2t right) left( zeta
      left( 1/2+it right) right) ^2-Psi left( 1,1/4+i/2t right)
      left( zeta left( 1/2+it right) right) ^2-8,zeta left( 2,
      1/2+it right) zeta left( 1/2+it right) +8, left( zeta left(
      1,1/2+it right) right) ^2$



      where $Psi(1,x)$ is the 1st derivative of the digamma function and $zeta(n,t$) is the n-th derivative of the Riemann zeta function.



      The function G(t) has an essential singularity inside the range



      $(5.56057204723220 , 5.56180573101957)$



      I believe that if there was a root of Zeta off the line then it would cause $G(t)<0$ for some $t>5.56...$ . Therefore, can the Riemann hypothesis be stated as the positivity of $G(t)$ for all real t > $5.56...$ ?



      enter image description here




      complex-analysis derivatives logarithms riemann-zeta laplacian




      share|cite|improve this question











      Let $Z left( t right) =rm e^i left( -i/2 left( it lnGamma
      left( 1/4+i/2t right) -it ln Gamma left( 1/4-i/2t right)
      right) -1/2,ln left( pi right) t right) zeta left( 1/2+
      it right) $



      be the Hardy Z function



      then define the integral of the logarithmic derivative of Z



      $R(t)=frac 1pi int
      !frac frac rm drm dtZ left( t right) Z left( t
      right) ,rm dt=frac -1/2,it lnGamma left( 1/4-i/2t right) +1/2,it lnGamma
      left( 1/4+i/2t right) +ln left( zeta left( 1/2+it right)
      right) -i/2ln left( pi right) tpi $



      then let



      $G(t)=- left( frac rm d^2rm dt^2R left( t right)
      right) ^-1=-8,frac left( zeta left( 1/2+it right)
      right) ^2pi Psi left( 1,1/4-i/2t right) left( zeta
      left( 1/2+it right) right) ^2-Psi left( 1,1/4+i/2t right)
      left( zeta left( 1/2+it right) right) ^2-8,zeta left( 2,
      1/2+it right) zeta left( 1/2+it right) +8, left( zeta left(
      1,1/2+it right) right) ^2$



      where $Psi(1,x)$ is the 1st derivative of the digamma function and $zeta(n,t$) is the n-th derivative of the Riemann zeta function.



      The function G(t) has an essential singularity inside the range



      $(5.56057204723220 , 5.56180573101957)$



      I believe that if there was a root of Zeta off the line then it would cause $G(t)<0$ for some $t>5.56...$ . Therefore, can the Riemann hypothesis be stated as the positivity of $G(t)$ for all real t > $5.56...$ ?



      enter image description here





      complex-analysis derivatives logarithms riemann-zeta laplacian





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