Number Theory

   

Intuitive Explanation of the Riemann Hypothesis

Authors: John Atwell Moody

Let \alpha be the unique \Gamma(2) invariant form on H with a pole of residue 1 at i\infty and one of residue -1 at 1. The ratio [\alpha: i\pi dtau] tends to 1 at the upper limit of [0,i\infty). Let \mu_{pm}:TxH->H be the action of multiplying by \sqrt{g} and {1\over{\sqrt{g}} for g in the connected real multiplicative group T. For each real c in (0,1) and each unitary character \omega, the form g^{2-2c}\omega(g)\mu_+^*(\alpha-\pi d\tau)\wedge \mu_-^*)\alpha-i\pi d\tau is exact if and only if \zeta(c+i\omega_0)=0 where \omega_0 is chosen such that \omega(g)=g^{i\omega_0}.

The rate of change of the magnitude is given by an integral involving a unitary character. Conjecturally the rate seminegative on the region 0
The form descends to the real projective line, it is locally meromorphic there with one pole and integrates to \pi e^{i\pi ({3\over 2}s + 1}. The value \zeta(s)=0 if and only if the integral along the arc from 0 to \infty not passing 1 is zero. This implies the arc passing through 1 equals a residue. We begin to relate the equality with the condition Re(s)=1/2.

Comments: 14 Pages.

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Submission history

[v1] 2018-04-20 03:14:35
[v2] 2018-04-20 14:12:33
[v3] 2018-04-21 05:08:23
[v4] 2018-04-23 14:20:11
[v5] 2018-04-24 16:07:40
[v6] 2018-04-29 20:28:18
[v7] 2018-05-05 14:16:48

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