(* 
      The Liouville λ - 
        function *) 

λ[
        p_] := (faclist = FactorInteger[p]; (-1)^
          Sum[faclist[[i, 2]], {i, Length[faclist]}])

f[x_] = Gamma[0, x]/x; 
    t[n_, p_] = n f[2 π n^2/p];

grand[p_, 
        s_] = (p/(2 π))^
          s Gamma[s]/(s - 1)^2 Zeta[4  s - 2]/Zeta[2  s - 1]

Residue[grand[p, s], {s, 1}]

Residue[grand[p, s], {s, 3/4}]

Plot[Abs[grand[1, 1 + I t]], {t, 7, 20}]

NIntegrate[Abs[grand[1, 1 + I t]], {t, 7, 20}]

NIntegrate[Abs[grand[1, 1 + I t]], {t, 7, 15}]

(* 
      Here are some integrals done over the integration contour (Subscript[C, 2])*)

Plot[Abs[grand[1, σ + 7 I]], {σ, 1/2, 1}]

NIntegrate[
      Abs[grand[1, σ + 7 I]], {σ, 1/2, 1}]

Plot[Abs[grand[1, 1/2 + I t]], {t, -7, 7}, 
      PlotRange -> All]

NIntegrate[Abs[grand[1, 1/2 + I t]], {t, -7, 7}]

(* In section 4,  
      we made use of the inequality (∑_(u = 1)^∞) 
              u Exp[-u^2/a]  ≤  
        a/2.   Here is a table of the two: *) 

usum[a_] := 
      N[(∑_(u = 1)^∞) u Exp[-u^2/a]]

Table[{usum[a], a/2}, {a,  .1, 3,  .1}]