Time complexity of Sieve of Eratosthenes
Write a sum for the number of operations of removing multiples of primes. How many operations to remove all multiples of 103 for example from an array of n bits? How many operations to remove all multiples of a prime number p? So you get a sum.

Then you need to know what the sum 1/1 + 1/2 + 1/3 + ... + 1/n is (roughly) and importantly why. You see that by adding 1/1 + (1/2 + 1/3) + (1/4 + 1/5 + 1/6 + 1/7) + (1/8 + 1/9 ... + 1/15) + ... and there is an obvious upper bound. 

Now instead of adding 1/k for $2^m ≤ k &lt; 2^{m+1}$ you add 1/k for only the primes k with $2^m ≤ k &lt; 2^{m+1}$. How many are there? What sum do you get now? And then you have a sum that looks almost exactly like the one you started with and gives you the result.