# Enquiry on an inequality involving the sum of the reciprocals of primes

Let $p_k$ denote the $k-th$ prime. Do there exist some constant $A>0$ such that for every sufficiently large $k$,

$$\sum_{p\leq p_k} \frac{1}{p} > B + \log\log p_k + \frac{A}{\log p_k}$$

where $B$ is the Mertens constant?

It is known that

$$\sum_{p\leq p_k} \frac{1}{p} = B + \log\log p_k + O\left(\frac{1}{\log p_k} \right)$$

but nothing seems to be known about the sign of the implicit constant in the Landau $O$-symbol.

• No. Mertens original paper has the opposite inequality with some terms plus 4/log(n+1), which means you can compute n_0 for which A=5 works for n bigger than n_0 for the opposite inequality. Also, there is oscillation (see Diamond and Pintz) which means your inequality with any positive A fails infinitely often. Gerhard "Goes This Way And That" Paseman, 2017.05.13. May 13, 2017 at 14:04
• @GerhardPaseman: why not posting this as an answer?
– Seva
May 13, 2017 at 16:11
• Because I don't know if the question has a typo. If the question is meant as asked, I may post an answer. Gerhard "Isn't Sure Of The Question" Paseman, 2017.05.13. May 13, 2017 at 23:08