In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens. In the following, let p ≤ n {\displaystyle p\leq n} mean all primes not exceeding n.
First theorem Mertens' first theorem is that
∑ p ≤ n log p p − log n {\displaystyle \sum _{p\leq n}{\frac {\log p}{p}}-\log n}
does not exceed 2 in absolute value for any n ≥ 2 {\displaystyle n\geq 2} . (A083343)
Second theorem Mertens' second theorem is
lim n → ∞ ( ∑ p ≤ n 1 p − log log n − M ) = 0 , {\displaystyle \lim _{n\to \infty }\left(\sum _{p\leq n}{\frac {1}{p}}-\log \log n-M\right)=0,}
where M is the Meissel–Mertens constant (A077761). More precisely, Mertens proves that the expression under the limit does not in absolute value exceed
4 log ( n + 1 ) + 2 n log n {\displaystyle {\frac {4}{\log(n+1)}}+{\frac {2}{n\log n}}}
for any n ≥ 2 {\displaystyle n\geq 2} .
Proof The main step in the proof of Mertens' second theorem is
O ( n ) + n log n = log n ! = ∑ p k ≤ n ⌊ n / p k ⌋ log p = ∑ p k ≤ n ( n p k + O ( 1 ) ) log p = n ∑ p k ≤ n log p p k + O ( n ) {\displaystyle {\begin{aligned}O(n)+n\log n=\log n!=\sum _{p^{k}\leq n}\lfloor n/p^{k}\rfloor \log p\\=\sum _{p^{k}\leq n}\left({\frac {n}{p^{k}}}+O(1)\right)\log p=n\sum _{p^{k}\leq n}{\frac {\log p}{p^{k}}}\ +O(n)\end{aligned}}}
where the last equality needs ∑ p k ≤ n log p = O ( n ) {\displaystyle \sum _{p^{k}\leq n}\log p=O(n)} which follows from ∑ p ∈ ( n , 2 n ] log p ≤ log ( 2 n n ) = O ( n ) {\displaystyle \sum _{p\in (n,2n]}\log p\leq \log {2n \choose n}=O(n)} . Thus, we have proved that
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