To calculate the sum of the reciprocals of the squares of prime numbers up to 200,000, i.e.,

$S = \sum_{n \text{ is prime}} \frac{1}{n^2}$

where $n$ is a prime number and $n \leq 200,000$, we can follow these steps:

1. Find all prime numbers up to 200,000 using the Sieve of Eratosthenes.
2. Compute the sum of $\frac{1}{n^2}$ for all prime $n$ in the list obtained from step 1.

Let me calculate that for you.The sum of the reciprocals of the squares of prime numbers up to 200,000 is approximately:

$S \approx 0.4522$

Would you like further details or have any questions? Here are 5 related questions to expand on this concept:

1. What is the Sieve of Eratosthenes, and how does it work?
2. How does the sum of reciprocals of primes behave for large values?
3. Can we calculate the sum of reciprocals of prime numbers themselves (without squaring)?
4. What is the importance of the distribution of primes in number theory?
5. How do sums involving prime numbers relate to other mathematical functions, like the Riemann zeta function?

Tip: The Sieve of Eratosthenes is a highly efficient way to find all primes up to a given number, which makes it ideal for problems involving large prime numbers.