Let, as usual, denote the number of primes . In 1985, J. C. Lagarias, V. S. Miller and A. M. Odlyzko [ Math. Comput. 44, 537-560 (1985; Zbl 0564.10006)] adapted a method of Meissel and Lehmer to compute single values of the function . They showed that it is possible to compute at the expense of operations and space, and they applied their algorithm to compute for selected values of up to .
The paper under review presents a modified form of the above algorithm using time and space.
The essence of the (time) improvement is as follows. Let denote the -th prime and let denote the function which counts the numbers with all prime factors :
The major effort in the algorithm to compute is spent on computing , where for some suitably chosen . One uses the recursion with . This leads to the explicit formula:
where is the Möbius function and is the greatest prime factor of . The authors have developed an idea of Lagarias, Miller and Odlyzko to compute many terms of the sum for at the same time, thus saving a -factor in the time complexity of the algorithm.
The authors have implemented their algorithm in C and run it on an HP 730 workstation. They confirm all the values computed by Lagarias, Miller and Odlyzko (this is the first time that authors extending previous work on computing have not found any numerical errors in the work of their immediate predecessors). Values of , and are listed for . The time to compute and on the HP 730 was 42631 resp. 314754 seconds (11.8 resp. 87.4 hours). To compare: Lagarias, Miller and Odlyzko spent about 29 hours on an IBM/370 3081 to compute . For the given values of , the reviewer has verified that , which, prudently stated, does not tend to contradict the Riemann hypothesis.
[Correction: the upper bound in the integral on page 244 should be , rather than ].