Is the zeta function a complex mapping onto the infinite sum of 1/j^s terms?

[Janitor]

I came across this at another website which usually does not delve into mathematics at all.

Where are the zeros of zeta of s?
G.F.B. Riemann has made a good guess;
They're all on the critical line, saith he,
And their density's one over 2 p log t.

This statement of Riemann's has been like a trigger,
And many good men, with vim and with vigour,
Have attempted to find, with mathematical rigour,
What happens to zeta as mod t gets bigger.

The efforts of Landau and Bohr and Cramer,
Littlewood, Hardy and Titchmarsh are there,
In spite of their effort and skill and finesse,
In locating the zeros there's been little success.

In 1914 G.H. Hardy did find,
An infinite number do lay on the line,
His theorem, however, won't rule out the case,
There might be a zero at some other place.

Oh, where are the zeros of zeta of s?
We must know exactly, we cannot just guess.
In order to strengthen the prime number theorem,
The integral's contour must never go near 'em.

Let P be the function p minus Li,
The order of P is not known for x high,
If square root of x times log x we could show,
Then Riemann's conjecture would surely be so.

Related to this is another enigma,
Concerning the Lindelöf function mu sigma.
Which measures the growth in the critical strip,
On the number of zeros it gives us a grip.

But nobody knows how this function behaves,
Convexity tells us it can have no waves,
Lindelöf said that the shape of its graph,
Is constant when sigma is more than one-half.

There's a moral to draw from this sad tale of woe,
which every young genius among you should know:
If you tackle a problem and seem to get stuck,
Use R.M.T., and you'll have better luck.


Words by Tom Apostol

 

[DeadWolfe]

I'm speachless.

 

[Janitor]

Then you'd better go get yourself a spech.

Here's a question that came to me today. Does the zeta function serve as a mapping from the complex plane (values of s) onto the complex plane (values of the infinite sum of 1/j^s terms)? I have a gut feeling that the answer is "yes" and also that proving it is either really simple or really difficult, but not sort of hard.

It is obviously not a 1:1 mapping since millions of values of s are known by computer calculation to map to (0,0).