Evaluating this sum. Is there a closed form?

Question

Trying to evaluate this sum:

$$ S=\sum_{n=1}^\infty \ln(p_n^2)K_1(\ln(p_n^2)). $$ Here $p_n$ is the nth twin prime and $K_1$ is the modified bessel function of the second kind.

So one should be summing over $p=3,5,5,7,11,13,17,19,...$ $(5$ is double counted$).$

I want to show that $S<B$ where $B$ is Brun's constant. $B\approx1.902.$

Answer 1

I would be shocked if this series had any kind of closed form.

However, you don't need a closed form to prove that it's less than Brun's constant.

Compare the reciprocal function and your function:

Plot[{1/x,Log[x^2] BesselK[1,Log[x^2] ]},{x,2,10}]

Since all the terms are positive and the series converges, it's clear that $S<B$.

Proving the inequality which is apparent from the plot is not hard and can be done using the known properties of Bessel functions.