Manipulate giving errors when I plot the product of a step function and a continuous function

Riemann’s prime counting function J (link) takes half-values at every jump-discontinuity. So, I define it thus:

Clear["Global`*"]; 
floor[e_] := Quiet[Check[Floor[e], Floor[FullSimplify[e]]]]; 
riemannJ[x_] :=
   With[{δ = 0.01}, 
     Piecewise[{
       {(Sum[(1/α)* PrimePi[(x - δ)^(1/α)],
        {α, 1, floor[Log[x - δ]/Log[2]]}] + Sum[(1/α)*
        PrimePi[(x + δ)^(1/α)], {α, 1, 
        floor[Log[x + δ]/Log[2]]}])/2,
        PrimePowerQ[x]},
       {Sum[(1/α)*PrimePi[x^(1/α)], {α, 1, 
       floor[Log[x]/Log[2]]}],
       True}
   }]];

Viewed as a Plot, the function J[x]*x^(-s - 1) produces a series of continuous lines that are discontinuous from each other at all points where PrimePowerQ[x] = True (plot shown with with s = -2):

Module[{s = -2}, 
 Plot[riemannJ[x]*x^(-s - 1), {x, 1, 10}, GridLines -> Automatic]
]

I have a couple of problems:

1. That error message: is Mathematica passing inaccurate values of x to Plot?

2. I’d like to use Manipulate to chart the curve for different values of s. But the plot now has a pink fill. The code is:

   Manipulate[
    Plot[riemannJ[x]*x^(-s - 1), {x, 1, 10}, GridLines -> Automatic], 
    {{s, -2, Style["Choose s", Larger, Bold]}, -5, 5, 1/2}, 
    ControlType -> Setter]
   

and the output:

I assume that this is the same error (as evidenced but the line below the plot), only now it shows the plot in pink.

How do I get around this?

Lastly:

3. Is there a way to ‘join up’ the line segments at the points of discontinuity?

In[1]:= PrimePowerQ[1.2]
During evaluation of In[1]:= PrimePowerQ::exact: Argument 1.2` in PrimePowerQ is not an exact number.
Out[1]= PrimePowerQ[1.2]