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In Crandall and Pomerance "Prime Numbers: A Computational Perspective" Second Ed., pp. 121 just before Eq. (3.1) it says: "The number of steps in the sieve of Eratosthenes is proportional to $\sum_{p\leq N} N/p$, where $p$ runs over primes."

However, in the sieve of Eratosthenes one only sieves with primes $p \leq \sqrt{N}$, so I would say the sum should be $\sum_{p\leq \sqrt{N}} N/p$. Now, except for a constant term, the sum turns out to be the same in the two cases, so for the purposes of the complexity estimate in the book, it makes no difference. Nevertheless, I'd say this constitutes a typo.

Could someone please verify this?

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up vote 2 down vote accepted

Carl Pomerance verified it and thanked me.

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