I asked this question at the math stack swap, but it didn't get traction. Always curious to know the answer.

Numerical evidence suggests that:

$$ lim_ {N to + infty} sum_ {n = 1} ^ N frac {1} {n} sum_ {k = 0} ^ n (-1) ^ k {n choose k } frac {1} {(k + 1) ^ {s}} = s $$

or equivalent

$$ lim_ {N to + infty} H left (N right) + sum _ {n = 1} ^ {N} left ({

frac {1} {n} sum _ {k = 1} ^ {n} { left (-1 right) ^ {k} {n choose k} frac {1

} { left (k + 1 right) ^ {s}}}} right) = s $$

with $ H (N) $ = the $ N $-th Harmonic Number.

Convergence is quite slow, but clearly goes faster for negatives $ s $. In addition, calculations for non-integer values of $ s $ require high precision parameters (I used Maple, bet / gp and ARB).

However, according to Mathematica, the series diverges according to the "harmonic series test", although $ s $ as an integer, it agrees on convergence.

**Does this series converge for $ s in mathbb {C} $ ?**

Some numerical results below:

```
s=0.5
0.497702121, N = 100
0.499804053, N = 1000
0.499905919, N = 2000
s=-3.1415926535897932385
-3.14160222, N = 100
-3.14159284, N = 1000
-3.14159272, N = 2000
s=2.3-2.1i
2.45310498 - 1.94063637i, N = 100
2.33501943 - 2.09308517i, N = 1000
2.31996958 - 2.09923503i, N = 2000
```