### Canzoni sull'ipotesi di Riemann

Inviato:

**03 gen 2018, 21:31**Cercando su internet qualcosa sulla teoria analitica dei numeri ho trovato queste, mi sono piaciute un sacco e penso che possano piacere anche a un po' di gente brava in TdN qui sul forum.

The Zeta Function Song (Sung to the tune of “Sweet Betsy from Pike”)

Where are the zeros of zeta of s?

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

They’re all on the critical line, said he,

And their density’s one over 2πlogt.

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 names of Landau and Bohr and Cramér,

And Hardy and Littlewood and Titchmarsh are there,

In spite of their efforts and skill and finesse,

In locating the zeros no one’s had 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,

That there might be a zero at some other place.

Let P be the function π minus li,

The order of P is not known for x high,

If square root of x times logx we could show,

Then Riemann’s conjecture would surely be so.

Related to this is another enigma,

Concerning the Lindelof function µ(σ)

Which measures the growth in the critical strip,

And 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, we cannot just guess,

In order to strengthen the prime number theorem,

The path of integration must not get too near’em.

Tom Apostol, Number Theory Conference, Caltech, June 1955

What Tom Apostol Didn’t Know

André Weil has bettered 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 good moral to draw from this long tale of woe

That every young genius among you should know:

If you tackle a problem and seem to get stuck,

Just take it mod p and you’ll have better luck.

Anonymous (Saunders Mac Lane?), Cambridge University, 1973

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 lie 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.

Saunders Mac Lane

PS: viene già suggerita una canzone, ma a me questi versi hanno fatto venire in mente questa, in caso qualcuno volesse provare a cantarle diversamente: https://www.youtube.com/watch?v=-X9PCxw7aKY.

The Zeta Function Song (Sung to the tune of “Sweet Betsy from Pike”)

Where are the zeros of zeta of s?

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

They’re all on the critical line, said he,

And their density’s one over 2πlogt.

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 names of Landau and Bohr and Cramér,

And Hardy and Littlewood and Titchmarsh are there,

In spite of their efforts and skill and finesse,

In locating the zeros no one’s had 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,

That there might be a zero at some other place.

Let P be the function π minus li,

The order of P is not known for x high,

If square root of x times logx we could show,

Then Riemann’s conjecture would surely be so.

Related to this is another enigma,

Concerning the Lindelof function µ(σ)

Which measures the growth in the critical strip,

And 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, we cannot just guess,

In order to strengthen the prime number theorem,

The path of integration must not get too near’em.

Tom Apostol, Number Theory Conference, Caltech, June 1955

What Tom Apostol Didn’t Know

André Weil has bettered 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 good moral to draw from this long tale of woe

That every young genius among you should know:

If you tackle a problem and seem to get stuck,

Just take it mod p and you’ll have better luck.

Anonymous (Saunders Mac Lane?), Cambridge University, 1973

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 lie 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.

Saunders Mac Lane

PS: viene già suggerita una canzone, ma a me questi versi hanno fatto venire in mente questa, in caso qualcuno volesse provare a cantarle diversamente: https://www.youtube.com/watch?v=-X9PCxw7aKY.