# Evaluating this sum. Is there a closed form?

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 $$S where $$B$$ is Brun's constant. $$B\approx1.902.$$

• A proof that it has (or doesn't have) a closed form would likely require deciding whether there are infinitely many twin primes. – Robert Israel Aug 23 '19 at 23:21

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.

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