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Edit 9.11.2018:

(*Mathematica divergent sum start*)
nn = 500;
f[t_] = D[RiemannSiegelTheta[t], t];
g2 = Plot[
  Sign[RiemannSiegelZ[t]]*
   Abs[Zeta[1/2 + I*t]*
      Total[Table[
        Total[MoebiusMu[Divisors[n]]/Divisors[n]^(1/2 + I*t - 1)]/
         n, {n, 1, nn}]]]/(f[t] + HarmonicNumber[nn]), {t, 0, 60}, 
  PlotStyle -> Thickness[0.004], ImageSize -> Large, 
  PlotRange -> {-4, 4}]
(*end*)

(*Mathematica analytic continuation start*)
f[t_] = D[RiemannSiegelTheta[t], t];
epsilon = 1/40;
g3 = Monitor[
  Plot[Sign[RiemannSiegelZ[t]]*
    Abs[Zeta[1/2 + I*t]/Zeta[1/2 + I*t + (1 + epsilon) - 1]*
       Zeta[(1 + epsilon)]]/(f[t] + 1/epsilon + EulerGamma), {t, 0, 
    60}, PlotStyle -> {Red, Thickness[0.004]}, ImageSize -> Large, 
   PlotRange -> {-4, 4}], n]

Show[{g2, g3}, ImageSize -> 1000]
(*code line: "Show[{g2, g3}, ImageSize -> 1000]" added by Roger L. Bagula*)
(*end*)

The graph above cross the x-axis at nontrivial Riemann zeta zeros.

Edit 9.11.2018:

(*Mathematica divergent sum start*)
nn = 500;
f[t_] = D[RiemannSiegelTheta[t], t];
g2 = Plot[
  Sign[RiemannSiegelZ[t]]*
   Abs[Zeta[1/2 + I*t]*
      Total[Table[
        Total[MoebiusMu[Divisors[n]]/Divisors[n]^(1/2 + I*t - 1)]/
         n, {n, 1, nn}]]]/(f[t] + HarmonicNumber[nn]), {t, 0, 60}, 
  PlotStyle -> Thickness[0.004], ImageSize -> Large, 
  PlotRange -> {-4, 4}]
(*end*)

(*Mathematica analytic continuation start*)
f[t_] = D[RiemannSiegelTheta[t], t];
epsilon = 1/40;
g3 = Monitor[
  Plot[Sign[RiemannSiegelZ[t]]*
    Abs[Zeta[1/2 + I*t]/Zeta[1/2 + I*t + (1 + epsilon) - 1]*
       Zeta[(1 + epsilon)]]/(f[t] + 1/epsilon + EulerGamma), {t, 0, 
    60}, PlotStyle -> {Red, Thickness[0.004]}, ImageSize -> Large, 
   PlotRange -> {-4, 4}], n]

Show[{g2, g3}, ImageSize -> 1000]
(*code line: "Show[{g2, g3}, ImageSize -> 1000]" added by Roger L. Bagula*)
(*end*)

The graph above cross the x-axis at nontrivial Riemann zeta zeros.

$$\sum\limits_{k=1}^{\infty}\sum\limits_{n=1}^{\infty} \frac{T(n,k)}{n^c \cdot k^s} = \sum\limits_{n=1}^{\infty} \frac{\lim\limits_{z \rightarrow s} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(z-1)}}}{n^c} = \frac{\zeta(s) \cdot \zeta(c)}{\zeta(c + s - 1)}$$$$\sum\limits_{k=1}^{\infty}\sum\limits_{n=1}^{\infty} \frac{T(n,k)}{n^c \cdot k^s} = \sum\limits_{n=1}^{\infty} \frac{\lim\limits_{z \rightarrow s} \zeta(z)\sum\limits_{d|n} \frac{\mu(d)}{d^{(z-1)}}}{n^c} = \frac{\zeta(s) \cdot \zeta(c)}{\zeta(c + s - 1)}$$