Riemann's work had put this question into a quite different form.
黎曼的研究工作，把这个问题带入了一个完全不同的形式。
He had defined a certain function of the complex numbers, the 'zeta-function'.
他定义了一个复函数，叫作"ζ函数"，
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.
误差始终能保持这样小，基本上就等价于这个命题：黎曼ζ函数的零点全都分布在平面的某条直线上。
This assertion had become known as the Riemann Hypothesis.
这个命题被称为黎曼猜想。
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，都认为这个猜想是成立的，但却没有人能够给出证明。
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，希尔伯特把它列为20世纪的第四个数学难题，有的时候还说它是数学中最重要的问题。
Hardy had bitten on it unsuccessfully for thirty years.
哈代被这个问题困扰了30年，仍未获得成功。
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.
这是数论的核心问题，并引出了一系列相关问题，艾伦选择了其中一个，作为自己的研究方向。
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.
如果不考虑黎曼的改进，只考虑那个原始命题，即素数的稀释与对数函数成比，那么在特定的范围内，它总是会高估素数的数量。
Common sense, or 'scientific induction', based on millions of examples, would suggest that this would always be so, for larger and larger numbers.
对几百万个数值进行归纳，随着范围越来越大，可以看出这个现象似乎总是成立的。