雙語暢銷書《艾倫圖靈傳》第4章:彼岸新星(50)
Little advance was made until 1859, when Riemann developed a new theoretical framework in which to consider the question.
It was his discovery that the calculus of the complex numbers could be used as a bridge between the fixed, discrete, determinate prime numbers on the one hand, and smooth functions like the logarithm—continuous, averaged-out quantities—on the other.
他發現可以引入複數注作爲橋樑，連接離散的素數，與平滑的連續函數，比如對數函數。He thereby arrived at a certain formula for the density of the primes, a refinement of the logarithm law that Gauss had noticed.
黎曼由此得到了一個素數密度的公式，對高斯發現的對數規律做了一些改進。
Even so, his formula was not exact, and nor was it proved.
但是這個公式仍然不準確，而且也無法證明。Riemann's formula ignored certain terms which he was unable to estimate.
黎曼的公式，忽略了某些無法估算的誤差。These error terms were only in 1896 proved to be small enough not to interfere with the main result, which now became the Prime Number Theorem, and which stated in a precise way that the primes thinned out like the logarithm—
在1896年，人們認爲這種誤差太小，不會影響主要結果，但現在要找的是素數的分佈規律，這是一個精確的規律——not just as a matter of observation, but proved to be so for ever and ever.
光有觀察是不夠的，還要證明它永遠有效。But the story did not end here. As far as the tables went it could be seen that the primes followed this logarithmic law quite amazingly closely.
但是，故事並未結束，從素數列表來看，其分佈規律與對數函數驚人地吻合，The error terms were not only small compared with the general logarithmic pattern; they were very small.
總體來說，誤差不僅很小，而且是非常非常小。But was this also true for the whole infinite range of numbers, beyond the reach of computation?
但問題是，對於整個無限的範圍來說，總能保持這樣小嗎？—and if so what was the reason for it?
如果是，那麼這是爲什麼？
