Limit does not exist or limit does exist? What does the unevaluated output mean?

Question

From this short Mathematica program while investigating the convergence of the Dirichlet series for the Möbius function:

Clear[a, b, s, x];
s = 1/2 + 100*I;
Limit[1 - Sum[1/a^s, {a, 2, x}] + 
  Sum[Sum[1/(a*b)^s, {a, 2, x}], {b, 2, x}], x -> Infinity]

I get the output:

where Out[186]= is left unevaluated.

Does that mean that the limit exists but it is not known in the form of standard functions?

Answer 1

First, we consider

 s = 1/2 + 100*I; Sum[1/a^s, {a, 2, x}]

-1 + HarmonicNumber[x, 1/2 + 100 I]

Second (The crucial tool is an assumption.),

Sum[Sum[1/(a*b)^s, {a, 2, x},Assumptions -> b \[Element] PositiveIntegers], {b, 2, x}]

(-1 + HarmonicNumber[x, 1/2 + 100 I])^2

At last,

Limit[1 + (-1 + HarmonicNumber[x, 1/2 + 100 I])^2 - (-1 + 
HarmonicNumber[x, 1/2 + 100 I]), x -> Infinity]

ComplexInfinity

Addition. The OP additionally requires a*b<=x. In fact, this is not any restriction because the limit as x->Infinity is taken. The sum of a double series does not depend on an exhaustion. Since

ComplexExpand[1/Abs[(a*b)^(\[Sigma] + I*t)]]

(Sqrt[a^2] Sqrt[b^2])^-\[Sigma] E^(t Arg[a b])

, the convergence depends only on \[Sigma]. It's more or less clear, that \[Sigma] should be greater than 1 to this end (I don't go into math deep.). Therefore,

Sum[Sum[1/(a*b)^\[Sigma], {a, 2, Infinity}, Assumptions -> \[Sigma] > 1 &&
 b \[Element] PositiveIntegers], {b, 2, Infinity}, Assumptions -> \[Sigma] > 1]

1 - 2 Zeta[\[Sigma]] + Zeta[\[Sigma]]^2

The same with Sum[1/a^s, {a, 2, x}]. Making use of Mathematica, we obtain the same answer as in MSE. The true convergence of a series is its absolute convergence. In view of it the series from the question are not absolutely convergent for s = 1/2 + 100*I.