[seqfan] Constant C = ?

[Tomasz Ordowski]

Hello everyone!

Let's start with the conjecture

Sum_{k=2..n} (log(k)/k - 1/pi(k)) = log(n) - C + o(1).

Note that log(x)/x - 1/pi(x) ~ 1/x, since log(x) - x/pi(x) ~ 1.
Cf. https://en.wikipedia.org/wiki/Legendre%27s_constant

Let's define (without proof of convergence) the constant
C = lim_{n->oo} (log(n) - Sum_{k=2..n} (log(k)/k - 1/pi(k))).
What is the approximate value of this limit?
Is C a new math constant?

Best,

Thomas