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We know that $$ \sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x) $$

where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy $$ \sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/x)O(1/\ln x). $$

Thus both these divergent series grow at the same rate. Mertens' theorem was proved without using the prime number theorem, some 25 years before PNT was proved. However from these two examples, we cannot conclude that

$$ \lim_{n \to \infty} \frac{p_n}{n\ln n} = 1 $$ otherwise Mertens' would have been the first to prove PNT. My question is - based on the above two series, what are the technical difficulties that prevent us from reaching the conclusion that $p_n/n\ln n = 1$. There may be counter examples with other series, so such conclusions may not be true in general. However I am not interested in the general case. Instead I am asking only in case of the sequence $1/n\ln n$ and $1/p_n$.
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Why Merten's Mertens could not prove the prime number theorem?

We know that $$ \sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x) $$

where $c_1$ is a constant. Again Merten,s Mertens' theorem says that the primes $p$ satisfy $$ \sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/x). $$

Thus both these divergent series grow at the same rate. Merten's Mertens' theorem was proved without using the prime number theorem, some 25 years before PNT was proved. However from these two examples, we cannot conclude that

$$ \lim_{n \to \infty} \frac{p_n}{n\ln n} = 1 $$ otherwise Merten's Mertens' would have been the first to prove PNT. My question is - based on the above two series, what are the technical difficulties that prevents prevent us from reaching the conclusion that $p_n/n\ln n = 1$. There may be counter examples with other series, so such conclusions may not be true in general. However I am not interested in the general case. Instead I am asking only in case of the sequence $1/n\ln n$ and $1/p_n$.
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Why Merten's could not prove the prime number theorem?

We know that $$ \sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x) $$

where $c_1$ is a constant. Again Merten,s theorem says that the primes $p$ satisfy $$ \sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/x). $$

Thus both these divergent series grow at the same rate. Merten's theorem was proved without using the prime number theorem, some 25 years before PNT was proved. However from these two examples, we cannot conclude that

$$ \lim_{n \to \infty} \frac{p_n}{n\ln n} = 1 $$ otherwise Merten's would have been the first to prove PNT. My question is - based on the above two series, what are the technical difficulties that prevents us from reaching the conclusion that $p_n/n\ln n = 1$. There may be counter examples with other series, so such conclusions may not be true in general. However I am not interested in the general case. Instead I am asking only in case of the sequence $1/n\ln n$ and $1/p_n$.