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]]>G.F.B. Riemann has made a good guess,

They're all on the critical line, stated he,

And their density's one over 2\pi log t.

This statement of Riemann's has been like 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 Cramer,

And Littlewood, Hardy and Titchmarsh are there,

In spite of their efforts and skill and finesse,

(In) locating the zeros there's been no success.

In 1914 G.H. Hardy did find,

An infinite number that lay on the line,

His theorem however won't rule out the case,

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 Lindelof 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,

Lindelof 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.

Andre 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.

[...]

So what fraction of zeros on the line will be found,

When mod t is kept below some given bound?

Does the fraction, whatever, stay bounded below,

As the bound on mod t is permitted to grow?

The efforts of Selberg did finally banish,

All fears that the fraction might possibly vanish.

It stays bounded below, which is just as it should,

But the bound he determined was not very good.

Norm Levinson managed to show, better yet,

At two-to-one odds it would be a good bet,

If over a zero you happen to trip,

It would be on the line and not just in the strip.

Levinson tried in a classical way,

Weil brought modular means into play.

Atiyah then left and Paul Cohen quit,

So now there's no proof at all that will fit.

But now we must study this matter anew,

Serre points out manifold things it makes true.

A medal might be the reward in this quest,

For Riemann's conjecture is surely the best.