Zeros of Riemann Zeta Function- Euler Product and Functional Equation

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#1
1. The problem statement, all variables and given/known data

Question

Use the functional equation to show that for :

a) that
b) Use the functional equation and the euler product to show that these are the only zeros of for . And conclude that the other zeros are all located in the critical strip: . Show that these are symmetric about

2. Relevant equations

Euler product: defined for

Functional equation:

where

Also have which we know has simple poles at

So from this we can see that the that gave poles for gives arise to the zeros of at so that's the trivial zeros done.

3. The attempt at a solution

From the Euler product define for we can see that does not vanish for .

I think to make the rest of the conclusions about the critical strip and being symmetrically distributed about I need to use the functional equation.

But I'm not sure what to do... I want to look where it is positive and negative I guess. But with and which are positive and negative at different ranges of I'm not really sure what to do.. any hint greatly appreciated.
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4 years ago
#2
Hi,

I'm not sure what bits of this you've done and haven't done. Your attempt at a solution looks like it's on the right lines, but it's a little confusingly ordered.

First thing you should do (as you have done) is show that for . Then, from the functional equation:

I believe you can deduce that for , .

I think the symmetry of zeroes arises from plugging into the functional equation and proving that the and functions are non-zero within a suitable range.
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