Its falsity would mean that the zeta-function did take the value zero at some point which was off the special line, in which case this point could be located by brute force, just by calculating enough values of the zeta-function.
说它是错误的也就是说，ζ 函数的某些零点并不在那条特定的直线上。这可以通过暴力计算来验证，只要计算出足够多的ζ函数的值。
This programme had already been started; indeed Riemann himself had located the first few zeroes and checked that they all lay on the special line.
这个计划已经启动了，黎曼亲自计算了最初的一些零点，并认定它们确实都排在一条线上。
In 1935-6, the Oxford mathematician E.C. Titchmarsh had used the punched-card equipment which was then used for the calculation of astronomical predictions to show that (in a certain precise sense) the first 104 zeroes of the zeta-function did all lie on the line.
1935年至1936年，牛津大学的数学家E.C.蒂施马奇用天文预测中使用的打孔卡片设备，证明了ζ 函数的前104个零点都在一条线上。
Alan's idea was essentially to examine the next few thousand or so in the hope of finding one off the line.
艾伦的想法是，要检验接下来的几千个零点，希望能找到一个不在线上的。
There were two aspects to the problem.
这里面存在两个问题。
Riemann's zeta-function was defined as the sum of an infinite number of terms, and although this sum could be re-expressed in many different ways, any attempt to evaluate it would in some way involve making an approximation.
首先，黎曼的ζ 函数的定义，是一个无限项的和式，尽管可以表示成一些其它的形式，但这些变形都要涉及到估算的问题。
It was for the mathematician to find a good approximation, and to prove that it was good: that the error involved was sufficiently small.
对数学家来说，需要找到一个好的估算方法，并证明它是好的，也就是误差足够小。
Such work did not involve computation with numbers, but required highly technical work with the calculus of complex numbers.
这个工作并不涉及算术，而是涉及到与复变函数有关的技术。
Titchmarsh had employed a certain approximation which, rather romantically, had been exhumed from Riemann's own papers at Gttingen where it had lain for seventy years.
蒂施马奇使用的估算方法，是从黎曼70年前在哥廷根的论文中挖掘出来的。
But for extending the calculation to thousands of new zeroes a fresh approximation was required; and this Alan set out to find and to justify.
但是，要想计算几千个新的零点，就需要新的估算方法。艾伦现在就要动身去寻找并证明。