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Little advance was made until 1859, when Riemann developed a new theoretical framework in which to consider the question.
直到1859年，这个问题一直没有新的进展。直到这一年，黎曼提出了一个新想法，
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.
他发现可以引入复数注作为桥梁，连接离散的素数，与平滑的连续函数，比如对数函数。
He thereby arrived at a certain formula for the density of the primes, a refinement of the logarithm law that Gauss had noticed.
黎曼由此得到了一个素数密度的公式，对高斯发现的对数规律做了一些改进。
Even so, his formula was not exact, and nor was it proved.
但是这个公式仍然不准确，而且也无法证明。
Riemann's formula ignored certain terms which he was unable to estimate.
黎曼的公式，忽略了某些无法估算的误差。
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，人们认为这种误差太小，不会影响主要结果，但现在要找的是素数的分布规律，这是一个精确的规律——
not just as a matter of observation, but proved to be so for ever and ever.
光有观察是不够的，还要证明它永远有效。
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，故事并未结束，从素数列表来看，其分布规律与对数函数惊人地吻合，
The error terms were not only small compared with the general logarithmic pattern; they were very small.
总体来说，误差不仅很小，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?
但问题是，对于整个无限的范围来说，总能保持这样小吗？
—and if so what was the reason for it?
如果是，那么这是为什么？
