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 Feb 26 awarded Yearling Feb 9 awarded Popular Question Sep 24 awarded Autobiographer Sep 24 awarded Nice Question Apr 11 awarded Nice Question Nov 26 comment Determining the number of classes @KCd: I am learning this material. I know some ring theory and galois theory. i know that the group of classes is finite and abelian and also know minkowski's theorem. i think embedding this in a vector space and using properties of ideals there is a way to solve this but I need help. how would you do it? Nov 25 asked Determining the number of classes Nov 5 comment Equivalence to the prime number theorem @GerryMyerson: The edits seem to be exactly what I am looking for. Assuming $\sum_{n=1}^x\mu(n)=o(x)$ is fine, as I know how to prove that from PNT. Can you post the solution? I think such a solution is worth the bounty and acceptance. Google books doesn't give me the good section. Nov 5 comment Equivalence to the prime number theorem @Phira: This is a good point, so I give you +1, but it is not exactly what I am looking for. Oct 30 revised Equivalence to the prime number theorem deleted 12 characters in body Oct 30 comment Equivalence to the prime number theorem I have put all my reputation for a bounty, and edited more. Oct 30 awarded Promoter Oct 30 revised Equivalence to the prime number theorem added 250 characters in body Oct 30 revised Equivalence to the prime number theorem deleted 19 characters in body Oct 29 awarded Critic Oct 29 comment Equivalence to the prime number theorem I already know this, this is not my question. What you have shown is that $$\sum_{n\leq x} \frac{\Lambda(n)}{n} =\log x +O(1)$$ where the constant must be less then $2$. This is one of mertens estimates. What I want is significantly stronger, and that is $$\sum_{n\leq x} \frac{\Lambda(n)}{n} =\log x -\gamma +o(1).$$ This last estimate implies the prime number theorem, so you cannot hope to prove it without using something that strong. Oct 29 revised Equivalence to the prime number theorem deleted 76 characters in body Oct 29 comment Equivalence to the prime number theorem @anon: you are correct. Oct 28 asked Equivalence to the prime number theorem Oct 1 awarded Teacher