## 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.

**Download:** **PDF**

### 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

**Unique-IP document downloads:** 211 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*