# Riemann Hypothesis equivalence

1. Oct 13, 2011

### zetafunction

let be the function $$\sum_{\rho} (\rho )^{-1} =Z$$

and let be the sum $$S= \sum_{\gamma}\frac{1}{1/4+ \gamma ^{2}}$$

here 'gamma' runs over the imaginary part of the Riemann Zeros

then is the Riemann Hypothesis equivalent to the assertion that $$S=2Z$$ ??

2. Oct 15, 2011

### jackmell

Yes. What happens if the RH is false? That means the conjugate zeros are off the critical line and symmetric to it (they come in four's in that case and not just two). How would that affect the two sums if that happened?

3. Oct 18, 2011

### lostcauses10x

In other words it is mirrored. If goes up in the uper half, it will be opposite in the bottom half.
The real line is the mirror line.

4. Oct 19, 2011

### zetafunction

i got the desired result in

however it seems to good to be true http://vixra.org/pdf/1110.0041v1.pdf

manage to prove that $$\sum_{t}(1/4+t ^{2})^{-1}=2+ \gamma -log(4\pi)$$

here 't' runs over the imaginary part of the Riemann Zeros, i have used the Riemann-Weil formula to prove it.

5. Oct 19, 2011

### lostcauses10x

zetafunction
What do you mean by Riemann Zeros?
Non trivial, trivial, or both??

6. Oct 19, 2011

### zetafunction

lostcauses10x ..

i mean the imaginary part of the zeros ON THE CRITICAL STRIP $$0<Re(s)<1$$

7. Oct 19, 2011

### lostcauses10x

zetafunction
thanks.

8. Oct 29, 2011

### lostcauses10x

Had to give this a bit of thought. Yet when examining the couture relations of the zeta function and the non trivial zeros a relation of the reflection property of the real line also shows up with the relation of the real line with limits due to the intersect of the non trivial zeros. A perpendicular intersect.

It seems to me this relation is directly proportional to the real line, and if the hypothesis is true, this relation is directly proportional and directly related tto the reflection and imaginary parts and perpendicular intersects at the real line.

Were as this limited reflective property appears around the non trivial zeros, were it starts on the real part contour is a bit of a problem to find: even if the hypothesis is true.

Just an observation, yet interesting result of what was said here. It does put the function in a bit better perspective to me. Of course this thought process is just beginning for me.