## リーマン予想の歌

### English

Where Are the Zeros of Zeta of S?
Words by Tom M. Apostol
(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, 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

### 日本語訳

ゼータ函数の零点はどこに？

("パイクから来た愛しのベッツィー"にのせて)

ゼータ函数の零点はどこにある？
G.F.B.リーマンは予想した

そしてその密度は$1/2 \pi \log t$

このリーマン予想は話題となった

tの絶対値が大きくなるとゼータに何が起こるかを

ランダウとボーアとクラメールの奮闘
ハーディとリトルウッドとティッチマーシュも居た

1914年にG.H.ハーディは見つけた

でもその定理では見い出せなかった

pを函数$\pi-Li$とする

もしそれがxの平方根掛ける$log x$だと示せれば
リーマンの予想は正しいと言える

これに関連した別の謎
リンデレーフの函数$\mu(\sigma)$

でもこの函数がどうふるまうかは誰にも分からない

リンデレーフ曰く そのグラフの形は
$\sigma$が1/2より大きければ定数になるのだとか

アンドレ・ヴェイユはリーマン予想を改良した
より素敵なゼータ函数を使って

もし問題に立ち向かって行き詰まってしまったら
pの法を取れば上手く行くかもしれない
7106137703271335173 https://www.storange.jp/2014/09/blog-post.html https://www.storange.jp/2014/09/blog-post.html リーマン予想の歌 2014-09-01T19:14:00+09:00 https://www.storange.jp/2014/09/blog-post.html Hideyuki Tabata 200 200 72 72