雙語暢銷書《艾倫圖靈傳》第4章:彼岸新星(51)
Riemann's work had put this question into a quite different form.
He had defined a certain function of the complex numbers, the 'zeta-function'.
他定義了一個複函數，叫作"ζ函數"，It could be shown that the assertion that the error terms remained so very smAll, was essentially equivalent to the assertion that this Riemann zeta-function took the value zero only at points which all lay on a certain line in the plane.
誤差始終能保持這樣小，基本上就等價於這個命題：黎曼ζ函數的零點全都分佈在平面的某條直線上。
This assertion had become known as the Riemann Hypothesis.
這個命題被稱爲黎曼猜想。Riemann had thought it Very likely' to be true, and so had many others, but no proof had been discovered.
黎曼本人，以及其他很多人，都認爲這個猜想是成立的，但卻沒有人能夠給出證明。In 1900 Hilbert had made it his Fourth Problem for twentieth century mathematics, and at other times called it 'the most important in mathematics, absolutely the most important'.
1900年，希爾伯特把它列爲20世紀的第四個數學難題，有的時候還說它是數學中最重要的問題。Hardy had bitten on it unsuccessfully for thirty years.
哈代被這個問題困擾了30年，仍未獲得成功。This was the central problem of the theory of numbers, but there was a constellation of related questions, one of which Alan picked for his own investigation.
這是數論的核心問題，並引出了一系列相關問題，艾倫選擇了其中一個，作爲自己的研究方向。The simple assumption that the primes thinned out like the logarithm, without Riemann's refinements to the formula, seemed always to overestimate the actual number of primes by a certain amount.
如果不考慮黎曼的改進，只考慮那個原始命題，即素數的稀釋與對數函數成比，那麼在特定的範圍內，它總是會高估素數的數量。Common sense, or 'scientific induction', based on millions of examples, would suggest that this would always be so, for larger and larger numbers.
對幾百萬個數值進行歸納，隨着範圍越來越大，可以看出這個現象似乎總是成立的。
