In the previous exercise we studied Legendre’s prime-counting function. In today’s exercise we will look at two more prime-counting functions, one from Ernst Meissel in the late 1800s and the other from Derrick Lehmer in the mid 1990s.

Both formulas are based on Legendre’s original formula; in both cases, they simply rearrange the terms of Legendre’s formula to eliminate some work. We won’t try to give derivations, as they are complex and chock-full of Greek letters; if you are interested, Hans Riesel’s book Prime Numbers and Computer Methods for Factorization was the source for all three formulas.

Meissel: \pi(x) = \phi(x, c) + \frac{(b+c-2)(b-c+1)}{2} - \sum_{i=c+1}^b \pi\left(\left\lfloor\frac{x}{p_i}\right\rfloor\right), where b = \pi(\lfloor \sqrt{x} \rfloor) and c = \pi(\lfloor \sqrt[3]{x} \rfloor)

Lehmer: \pi(x) = \phi(x, a) + \frac{(b+a-2)(b-a+1)}{2} - \sum_{i=a+1}^b \pi \left( \left\lfloor \frac{x}{p_i} \right\rfloor \right) - \sum_{i=a+1}^c \sum_{j=i}^{b_i} \left( \pi \left( \left\lfloor \frac{x}{p_i p_j} \right\rfloor \right) - (j-1) \right), where a = \pi(\lfloor \sqrt[4]{x} \rfloor), b = \pi(\lfloor \sqrt{x} \rfloor), c = \pi(\lfloor \sqrt[3]{x} \rfloor), and b_i = \pi(\lfloor \sqrt{x / p_i} \rfloor) for a < i <= c.

Your task is to implement the prime-counting functions of Meissel and Lehmer, then compare timings with Legendre's prime-counting function of the previous exercise. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


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