A thought about the Riemann hypothesis

[r731]

This is the Riemann Zeta function $Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$. It equals 0 only at the negative integers on the real axis and numbers of form $1/2+x i$.

The series can be expanded to this:

$$\sum_{n=1}^{\infty} \frac{1}{n^s} = \sum_{n=1}^{\infty} \frac{1}{n^{1/2 + xi}} = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}n^{xi}} = 0$$

I'm not sure what theorem (from real analysis) to apply to proceed.

<Moderator's note: please use the proper wrappers, ## ## and $$ $$, for LaTeX.>

 

[martinbn]

To proceed to what? What are you trying to do? To prove the hypothesis!

 

[r731]

If the equation is satisfied for all $x\in R$ then the hypothesis is true. Rudin's book has useful theorems about infinite series.

 

[mfb]

This is the Riemann Zeta function $Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$.

That formula only works in a limited range, and only in places with trivial zeros. It's not helping at all in the interesting region where you need the complex continuation. That makes everything that follows irrelevant.

 

[Office_Shredder]

If the equation is satisfied for all $x\in R$ then the hypothesis is true. Rudin's book has useful theorems about infinite series.

I think this is false

1.) The hypothesis is there are only zeroes with real -1/2. All you are doing is computing zeroes with that imaginary part, not disproving the existence of other zeroes

2.) It's not identically zero when the real part is -1/2. Your formula is simply not true for all x.

3.) I don't think the infinite series converges that you have written down, so you have to find another representation of the function in that region to do any math anyway.

 

[r731]

$s=1/2 + xi$ represent the complex numbers with real part 1/2. The infinite series is the zeta function. It should output zero when the real part is 1/2 (for all imaginary parts).

3.) I don't think the infinite series converges that you have written down, so you have to find another representation of the function in that region to do any math anyway.

The infinite series is the zeta function.

 

[phyzguy]

The infinite series is the zeta function.

No, it's not, as has already been pointed out to you. The Riemann zeta function is the analytic continuation of that series.

 

[martinbn]

$s=1/2 + xi$ represent the complex numbers with real part 1/2. The infinite series is the zeta function. It should output zero when the real part is 1/2 (for all imaginary parts).

It this were true, the it would be identically zero.