I want to see more of this constant

Number Theory Level 5

$\Large \lim_{n \to \infty} \left( \sum_{p\leq n}\frac{1}{p}-\ln(\ln(n))\right)$

If the above limit equals $M$ , find $\lfloor100M \rfloor$ .

Clarification : The sum is over all primes less than or equal to $n$ .

The answer is 26.

1 solution

Mark Hennings
Jan 21, 2016

$M$ is the Meissel-Mertens constant: $M \,=\, 0.2614972128476427837554268386086958590516...$ . This result is the formula that defines $M$ . Without reproducing Merten's Second Theorem, there is nothing more to prove.

On the other hand, the result is interesting, since it is a stepping-stone to the Prime Number Theorem.

I want to see more of this constant

The answer is 26.

1 solution

0 pending reports