Quantum Physics

   

The Laplacian of The Integral Of The Logarithmic Derivative of the Riemann-Siegel-Hardy Z-function

Authors: Stephen Crowley

The integral R(t)=π^(-1) (lnζ(1/2+i t)+iϑ(t)) of the logarithmic derivative of the Hardy Z function Z(t)=e^(i ϑ(t)) ζ(1/2+i t), where ϑ(t) is the Riemann-Siegel vartheta function, and ζ(t) is the Riemann zeta function, is used as a basis for the construction of a pair of transcendental entire functions ν(t)=-ν(1-t)=-(ΔR(i/2-i t))^(-1)=- G(i/2-i t) where G=-(ΔR(t))^(-1) is the derivative of the additive inverse of the reciprocal of the Laplacian of R(t) and χ(t)=-χ(1-t)=ν˙(t)=-i H(i/2-i t) where H(t)=G˙(t) has roots at the local minima and maxima of G(t). When H(t) and H˙(t)=G¨(t)=ΔG(t)>0, the point t marks a minimum of G(t) where it coincides with a Riemann zero, i.e., ζ(1/2+i t)=0, otherwise when H(t)^=0 and H˙(t)=ΔG(t)<0, the point t marks a local maximum G(t), marking midway points between consecutive minima. Considered as a sequence of distributions or wave functions, ν_n(t)=ν(1+2n+2t) converges to ν_∞(t)=lim_(n→∞)ν_n(t)=sin^2(π t) and χ_n(t)=χ(1+2n+2t) to χ_∞(t)=lim_(n→∞)χ_n(t)=-8 cos(π t)sin(π t)

Comments: 11 Pages.

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Submission history

[v1] 2015-10-28 23:22:39
[v2] 2015-10-29 17:45:02
[v3] 2015-10-30 01:54:27
[v4] 2015-11-04 17:47:29
[v5] 2015-11-08 17:19:36
[v6] 2015-11-10 18:05:36
[v7] 2015-11-21 16:33:31
[v8] 2016-01-31 18:59:27
[v9] 2016-02-01 13:43:55
[vA] 2016-02-26 19:52:02
[vB] 2016-03-27 01:08:58

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