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On page 66, Riesel correctly derives formula 3.8 however the jump to formula 3.9 is unjustified according to us.
1) Prime counting should be independent on the prime sieve used.
2) Although primes between sqrt(x) and x do not contribute to x being prime, they are an important part in the total probability formula and should not be omitted in the formula.
3) The formula for total probability is at the origin from which the formula for the individual probability is derived. So, we can not say that from going from the individual to the total probability, we forgot to take dependence into account.
For more information:
http://www.slideshare.net/ChrisDeCorte1/why-i-believe-that-riesel-is-wrong-about-prime-probabilities
http://www.slideshare.net/ChrisDeCorte1/probabilistic-approach-to-prime-counting
 

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