黎曼假设之歌：Zeta函数的零点在哪里？
这两天我在瘾Malloc新买的《素数之恋》，看到附录里面有一个非常有意思的“周边”，便拿过来和大家分享一下。歌曲的名称叫做”Where are the zeros of zeta of s ?“，这首歌的歌词是加利福尼亚理工学院的荣誉退休数学教授Tom Apostol在1955年写的（大概是目前为止我发过的最火星的东西了）。这首歌会让我想起Matrix67的“史上最牛诗歌”，我想我这个应该也不差吧～
Where are the zeros of zeta of s?
to the tune of “Sweet Betsy from Pike”; words by Tom Apostol
Where are the zeros of zeta of s?
G.F.B. Riemann has made a good guess,
They’re all on the critical line, sai he,
And their density’s one over 2pi log t *.
This statement of Riemann’s has been like trigger
And many good men, with vim and with vigor,
Have attempte 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, we cannot just guess,
In orer to strengthen the prime number theorem,
The integral’s contour must not get too near ‘em.
注释*：这里的2pi log t直接复制了我找歌词的原文。但是在素数之恋上面的版本是pi/2 *log T，有不同的版本……这个式子的含义是实部为1/2的那条直线上的零点密度，即高度为T以内的零点在单位长度内的个数。
歌曲使用的是《Sweet Betsy from Pike》的旋律，这个旋律第一次出现是在19世纪中流行的英国歌曲《Villikens and his Dinah》中。
这里提供这首歌的mp3甚至mpeg下载，都不大，但是并不快。
歌词来源：http://www.math.wisc.edu/~robbin/funnysongs.html 。BTW，这个网站上面还有很多”funny songs”的歌词。
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标签：Geek, Zeta, 未解决, 黎曼假设