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Where are the zeros of zeta of s?
G.F.B. Riemann has made a good guess:
«They're all on the critical line,» stated he,
«And their density's one over two pi log T».

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

The efforts of Landau and Bohr and Cramér,
Hardy and Littlewood and Titchmarsh are there.
In spite of their effort and skill and finesse,
In locating the zeros there's been no success.

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

Let P be the function pi 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.

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

André Weil has improved on old Riemann's fine guess
By using a fancier zeta of s.
He proves that the zeros are where they should be,
Provided the characteristic is p.

There's a moral to draw from this long tale of woe
That every young genius among you must know:
If you tackle a problem and seem to get stuck,
Just take it mod p and you'll have better luck.

Примечания